Patterns described by Discrete and
Continuous Dynamical systems
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus Dr. D. D. Breimer,
hoogleraar in de faculteit der Wiskunde en
Natuurwetenschappen en die de Geneeskunde,
volgens besluit van het College voor Promoties
te verdedigen op woensdag 9 juni 2004
klokke 15.15 uur
door
José Antonio Rodrı́guez
geboren te Baracaldo (Spanje)
op 23 januari 1976
Samenstelling van de promotiecommisie:
promotor:
prof. dr. ir. L.A. Peletier
referent:
prof. dr. A.R. Champneys (University of Bristol)
overige leden:
prof. dr. S. Verduyn Lunel
prof. dr. R. van der Hout
Dr. V. Rottsschäfer
Dr. R.C.A.M. van der Vorst (Vrije Universiteit Amsterdam)
The universe cannot be read until we have learned the language and become
familiar with the characters in which it is written. It is written in mathematical
language.
Galileo Galilei
If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
Richard Feynman
A mi familia, Begoña, Mari Carmen, Antonio y Patxi.
Thomas Stieltjes Institute for Mathematics
Contents
Preface
7
1 Introduction
1.1 Classical second order differential equations . . . .
1.2 Fourth order differential equations . . . . . . . . .
1.2.1 Linearisation around the uniform equilibria
1.2.2 Embedded solitons . . . . . . . . . . . . . .
1.2.3 Analytical methods . . . . . . . . . . . . . .
1.3 Pattern selection in the SH equation . . . . . . . .
1.3.1 Gradient systems . . . . . . . . . . . . . . .
1.3.2 The continuous model . . . . . . . . . . . .
1.3.3 The discrete model . . . . . . . . . . . . . .
1.4 Travelling waves on a lattice . . . . . . . . . . . . .
1.4.1 Lattices . . . . . . . . . . . . . . . . . . . .
1.4.2 Fronts on a lattice . . . . . . . . . . . . . .
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9
9
13
15
17
20
24
24
25
26
36
36
36
2 Embedded solitons
2.1 Introduction . . . . . . . . . . . . . . .
2.2 Preliminaries . . . . . . . . . . . . . .
2.3 Existence of homoclinic orbits . . . . .
2.4 Asymptotics . . . . . . . . . . . . . . .
2.5 Homoclinic orbits to periodic solutions
2.6 The extended bifurcation branch . . .
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41
41
45
48
55
58
63
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3 Fronts on a lattice
69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 A lower bound for the wave speed c . . . . . . . . . . . . . . . . 74
5
3.3
3.4
3.5
3.6
3.7
3.8
4 The
4.1
4.2
4.3
4.4
4.5
4.6
Outline . . . . . . . . . . . . . . . . . . . .
The characteristic equation . . . . . . . . .
The expansion of the fundamental solution
Monotonicity . . . . . . . . . . . . . . . . .
The final solution . . . . . . . . . . . . . . .
The range c < c0 . . . . . . . . . . . . . . .
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76
79
82
86
89
92
discrete Swift-Hohenberg equation
Introduction . . . . . . . . . . . . . . . . . . .
Preliminaries . . . . . . . . . . . . . . . . . .
The discrete Fisher-Kolmogorov equation . .
4.3.1 Symmetric solutions . . . . . . . . . .
4.3.2 Antisymmetric solutions . . . . . . . .
4.3.3 Nonsymmetric equilibria . . . . . . . .
The discrete Swift-Hohenberg equation . . . .
4.4.1 Symmetric dynamics . . . . . . . . . .
Dynamics . . . . . . . . . . . . . . . . . . . .
Phase plane analysis and large time behavior
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103
103
105
110
111
115
117
118
119
128
132
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at the origin
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139
139
140
144
144
A Appendix
A.1 Proof of Lemma 2.3 . . . . .
A.2 Weak convergence . . . . . .
A.3 Values of qk , ζk and of uk and
A.4 Proof of Theorem 4.1 . . . . .
. .
. .
u00k
. .
Bibliography
148
Samenvatting
155
Afterword
157
Curriculum vitæ
159
6
Preface
The subject of this thesis is the mathematical study of some special differential
equations arising from different fields of science. In the introductory chapter
we give a summary of the main results of the book. The other three chapters
correspond to three papers, written together with prof. dr. ir. L.A. Peletier.
Chapter 2 corresponds to the paper Homoclinic orbits to a saddle-center in a
fourth order differential equation, and it has been accepted for publication in the
Journal of Differential Equations. Chapter 3 corresponds to the paper Fronts on
a lattice, and was accepted by the journal Differential and Integral Equations.
Chapter 4 will be the content of another mathematical article to be submitted
in the near future, The discrete Swift-Hohenberg equation.
7
8
Chapter 1
Introduction
1.1
Classical second order differential equations
In the mathematical description of natural phenomena, a prominent role is
played by differential equations already introduced in the times of Isaac Newton
and Gottfried Wilhelm Leibnitz. Second order differential equations have been
long studied since the formulation by Newton of the second law of motion.
In the field of partial differential equations (PDE’s), second order linear
differential equations have been taken as models for the study of many physical
phenomena, and some of them have special names like the Laplace equation, the
wave equation and the heat equation.
These three equations can be written as
Lt u − ∆u = 0,
(1.1)
where Lt is a linear differential operator with respect to t, with constant coeffi∂2
cients. In the Laplace equation, Lt = 0, in the wave equation, Lt = ∂t
2 and in
∂
the heat equation, Lt = ∂t
.
The three equations have something in common: they are linear, which
means that if one has two solutions of the problem, then the sum of them, or
any number of them, multiplied by a constant, is also a solution. They are also
autonomous, which means that the differential equation or expressions explicitly
depend on the unknown u, but not on time or space.
While these equations have been able to explain important physical phenomena or have been taken as models for different processes, in recent decades,
9
particularly since the seventies, equations of a different type have been considered to describe phenomena that are shown to be impossible for equations of
second order.
There exist many different generalizations of those second order equations
and we do not attempt here to cover them all, not even the most important
ones. It is also not our intention to investigate different general methods or approaches. We are more interested in studying particular equations arising from
different applications from physics (mainly) and try to study specific questions,
like existence, uniqueness, stability and, specially, the qualitative properties of
certain types of solutions, considered of particular interest. Some of those types
are equilibria, travelling waves, periodic orbits, etc.
A slight modification of equation (1.1) is the following:
∂u
− ∆u = f (u).
∂t
(1.2)
Note that equation (1.2) looks similar to the heat equation, but we have added
the nonlinear function f . This equation is called a reaction-diffusion equation:
the Laplacian term corresponds to diffusion and the function f corresponds to
reaction. Note that roots of f correspond to homogeneus equilibria of equation
(1.2). Typical choices for f are the parabolic function f (u) = u − u2 . Then,
equation (1.2) is called a monostable equation. If f is given by the cubic function
f (u) = u − u3 , then equation (1.2) is called the bistable Fisher-Kolmogorov
equation.
When we seek equilibria of equation (1.2), they must satisfy the equation
−∆u = f (u).
(1.3)
Suppose that we consider equation (1.3) in one dimension, i.e. x ∈ . Then,
phase-plane analysis can help us to describe completely all the equilibria. It can
be rewritten as a second order differential system
du
dx
dv
dx
= v,
= −f (u).
Let us define the Energy functional
def
E(u, v) =
1 2
v (x) +
2
10
Z
u(x)
f (s) ds.
0
(1.4)
Then, we find that the Energy is constant along orbits, i.e.
dE
= 0.
dx
Thus, when the initial data (u0 , v0 ) is fixed, the orbit is a subset of the graph of
def
E(u, v) = E0 = E(u0 , v0 ). Therefore, considering different values for E0 ∈ will
give us the different possibilities for orbits. In the next plot, we show the phaseplane picture for the parabolic and the cubic nonlinearities. We draw bounded
orbits with thin solid lines (corresponding with periodic orbits), unbounded
orbits with dashed lines, and with thick lines the orbits that separate them. Note
that for the cubic nonlinearity, two heteroclinic orbits connecting the steady
states (u, v) = (±1, 0) separate the bounded and unbounded orbits, while for
the quadratic nonlinearity, a homoclinic orbit to the steady state (u, v) = (1, 0)
makes the separation.
2.5
1.5
2
1
1.5
1
0.5
0
v
v
0.5
0
−0.5
−0.5
−1
−1.5
−1
−2
−2.5
−2
−1.5
−1
−0.5
0
u
0.5
1
1.5
2
(a) f (u) = u − u3
−1.5
−1
−0.5
0
0.5
u
1
1.5
2
(b) f (u) = u − u2
Figure 1.1: Phase plane plot for different nonlinearities
The periodic orbits are uniquely determined by the Energy. In both cases
we have that there exist two critical values E − < E + ∈
such that for every
E ∈ (E − , E + ) there exists a unique periodic orbit solving the equation. In the
cubic case, E − = 0 and E + = 14 , while in the quadratic case, E − = 0 and
E + = 16 .
11
Apart from the Hamiltonian structure, we can define the Action as follows:
Z 1 0 2
def
J(u) =
u (x) − F u(x)
dx,
(1.5)
2
where
def
F (u) =
Z
u
f (s)ds.
(1.6)
0
In the definition of the potential F given by equation (1.6), one of the extrema
of the interval is u and the other one is chosen 0, but other choices are also
possible. The choice for 0 is the most convenient when studying solitons to the
origin, as we do in the first chapter. Therefore, this convention is the one we
will adopt; otherwise, we will specify an alternative definition. In any case, note
that the difference between any two of them is constant and in the variational
approach, we search for critical points of the action defined in equation (1.5).
Thus, the choice for the lower extremum plays no role in the discussion.
We will not describe in detail the appropriate space on which we define this
functional. Plainly, it depends on the domain that is being considered and the
type of solution being studied. Suppose that the domain is the whole real line,
and we want to find an orbit connecting two equilibria: the usual procedure is
then to find an appropriate Hilbert space H where we include the function and
the derivatives (a subspace of W 1,2 ) and consider an affine space of the type
φ(x) + H, where usually φ is an appropriate function connecting the equilibria.
The importance of this approach is that the Action is a Lyapunov function of
the parabolic PDE. This has important consequences on the large time behavior
of the PDE: for a big set of initial conditions, it converges to an equilibrium or
an orbit connecting them. That is the strongest motivation to study equilibria
of the PDE in detail.
Let us introduce an additional tool to study nonlinear parabolic partial differential equations like the Fisher-Kolmogorov equation, with different possibilities
for the nonlinearity function f : a Maximum Principle. Consider the Cauchy
problem for the partial differential equation (1.2) on some domain Ω. Denote
def
the solution by u(t, x) and let v(x) = u(0, x) be given. Consider v1 and v2 two
different initial conditions with v1 (x) 6 v2 (x) for every x ∈ Ω and let u1 (t, x)
and u2 (t, x) be their corresponding solutions for t > 0. Then a well known
Maximum Principle states that u1 (t, x) 6 u2 (t, x) for every t > 0 and x ∈ Ω.
The Hamiltonian structure, the variational structure and the maximum principle are basic tools that are present in Fisher-Kolmogorov type of equations
and are used a great deal in the study of nonlinear parabolic partial differential
12
equations. The combination of the three is also very specific for them and it
determines the type of orbits we can find.
1.2
Fourth order differential equations
In this and next section we want to study nonlinear fourth order differential
equations and in the last one we will study an example of a lattice differential
equation. We study them because they keep some of the structures that are
present in the previous equations, but at the same time we want to generalize
them. This will cause the drop of some of the main properties (the maximum
principle, for instance) and features and at the same time allow a larger variety
of orbits that can be expected. We will not attempt to cover all, not even
a large set of possible generalizations of the Fisher-Kolmogorov equation but
study some specific equations and show different behavior that we can find in
them.
The nonlinear parabolic partial differential equations have interesting properties and have been long studied, but they have also a relatively high number of
main axioms that limits the behavior of possible orbits. To study dynamical systems that exhibit behavior of higher complexity, more complex patterns, we are
forced to consider fourth order differential equations. Two such equations arising from Physics motivate our study, namely the Extended Fisher-Kolmogorov
equation (1.8) and the Swift-Hohenberg equation (1.9), defined below. As we
shall see, both equations possess a rich structure of solutions that can not be
found in nonlinear parabolic partial differential equations. This task has been
a big challenge in recent decades and new analytical tools are being developed
to study this type of equations.
Let us start by a canonical formulation of an equation of fourth order in
space and time.
∂4u
∂2u
∂u
= −γ 4 + β 2 − f (u),
∂t
∂x
∂x
t > 0,
x ∈ Ω.
(1.7)
Here, we consider the parameters γ > 0 and β ∈ .
The introduction of this canonical equation is motivated by two well known
equations. First, the Extended Fisher-Kolmogorov equation
∂u
∂4u ∂2u
= −γ 4 + 2 − f (u)
∂t
∂x
∂x
13
t > 0,
x ∈ Ω.
(1.8)
This is a particular case of equation (1.7) with β = 1. Note that for γ = 0, after
rescaling space, equation (1.8) becomes (1.2) , so it can be viewed as the “old”
4
Fisher-Kolmogorov equation (1.2) with the fourth order term −γ ∂∂xu4 added to
it. This explains the origin of its name.
Another model is the so called Swift-Hohenberg equation, which is usually
written as follows
2
∂u
∂2
=− 1+ 2
u + αu − u3 ,
t > 0, x ∈ Ω.
(1.9)
∂t
∂x
In contrast with equation (1.8), we can not drop the fourth order term in
(1.9) because the corresponding problem would be ill posed. While in the first
case, the procedure to follow is to extend the known results from the parabolic
equation (1.2) to equation (1.8), in the second case this procedure can not be
applied to equation (1.9).
Note that for α > 1, equation (1.9) fits into the class (1.7): we rescale
variables
√
1
s = (α − 1)t,
y = (α − 1) 4 x,
(1.10)
u = α − 1v
and then the variable v verifies equation (1.7) with γ = 1 and β = −
and the nonlinearity function
f (v) = v 3 − v.
√2
α−1
< 0,
(1.11)
For α < 1, equation (1.9) also fits into the class (1.7): we rescale variables
√
1
u = 1−αv
(1.12)
s = (1 − α)t,
y = (1 − α) 4 x,
and then the variable v verifies equation (1.7) with γ = 1 and β = −
and the nonlinearity function
f (v) = v + v 3 .
√2
1−α
< 0,
(1.13)
Note that both equations (1.8) and (1.9) fit into the class (1.7); but in the
first case, the parameter β > 0 while in the second one, β < 0. This makes a
relevant difference.
In both cases, the equilibria of equation (1.7) correspond to solutions of the
fourth order differential equation
γ
∂2u
∂4u
−
β
+ f (u) = 0,
∂x4
∂x2
14
x ∈ Ω.
(1.14)
As in the previous section, we introduce here some of the tools that are
used to study the dynamics of equation (1.14). First, we introduce the Energy
functional associated to it.
def
E(u) = −γ
du d3 u γ
+
dx dx3
2
d2 u
dx2
2
+
β
2
du
dx
2
− F (u),
(1.15)
where the potential function F is defined
as in (1.6). As in the previous sec
tion, we have that the Energy E u(x) is constant along the orbits that satisfy
equation (1.14).
Second, we introduce the Lagrangian action
)
2
Z ( 2 2
β du
γ d u
def
+ F u(x)
+
dx.
(1.16)
J(u) =
2 dx2
2 dx
Ω
Critical points of the Lagrangian action in the appropriate space correspond to
solutions of the differential equation (1.14). The domain of integration depends
on the sort of solution being studied.
Equation (1.14) contains two parameters γ and β and holds on x ∈ Ω.
However, when we consider the unbounded domain Ω = , we can scale one
parameter out. There exist several possibilities, but in the next chapter we use
the following procedure: we define
def
q = −
β
,
γ2
def
A = γ −1/4
and
y = Ax,
(1.17)
def
and we define the function v(y) = u(x). Then, v satisfies the following differential equation
d4 v
d2 v
(1.18)
+
q
+ f (v) = 0,
y∈ .
dy 4
dy 2
The differential equation (1.18) will be the starting point for our study in the
next chapter.
1.2.1
Linearisation around the uniform equilibria
Later in this chapter we will discuss the dynamics of equations (1.8) and (1.9).
Let us mention that both of them can be included in a broader class of dynamical systems called Gradient systems. As we shall discuss later, if the equilibria
15
are isolated, then the dynamics of the flow tend to an equilibrium. Thus, equilibria are particularly interesting solutions to be studied since they describe the
asymptotic behavior of the dynamics, besides the interest that they have on
their own.
Let us study the local behavior of the uniform equilibria, i.e., the solutions of
equations (1.8) and (1.9) which do not depend on x. They are given by the zeros
of the nonlinearity f . First, for the Extended Fisher-Kolmogorov equation, with
the nonlinearity (1.11), they are given by u = ±1 and u = 0. The spectrum
around u = ±1 is shown in the next sequence of pictures in Figure 1.2.
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−2
−1
0
(a) 0 < γ <
1
2
3
1
8
−3
−3
−2
−1
0
(b) γ >
1
2
3
1
8
Figure 1.2: Spectrum for the equation (1.8) around u = ±1
The spectrum depends on the parameter γ and two different situations are
possible. In the first case, the spectrum consists of four real eigenvalues, two
positive and two negative, and in the second case, four complex eigenvalues, one
in each quadrant. In the first case, the equilibrium is called a real saddle, and
in the second case, a saddle-focus. Note that in both cases, the spectrum is far
from the imaginary axis.
In contrast, the computation of the spectrum of the linearised operator
around u = 0 gives us a different picture, as in Figure 1.3(c). The spectrum
consists of two purely imaginary eigenvalues and two real ones of opposite sign.
In this case, the equilibrium is called a saddle-center.
Let us make similar calculations to compute the spectrum of the uniform
equilibria of the SH equation (1.9). Recall that the uniform equilibria u = ±1
exist for α > 1 while u = 0 is an equilibrium for every α ∈ . For u = ±1 and
α > 1, the location of the eigenvalues of the linearised operator is shown in the
16
sequence of pictures in Figure 1.3. Depending on the value of the parameter α,
the equilibrium u = ±1 is either a saddle-focus (for 1 < α < 3), a center (for
3 < α < 4) or a saddle-center (for α > 4) respectively.
3
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
−2
−3
−3
−2
−2
−1
0
1
2
(a) A saddle-focus for
α ∈ (1, 3)
3
−3
−3
−2
−2
−1
0
1
2
(b) A center for α ∈
(3, 4)
3
−3
−3
−2
−1
0
1
2
3
(c) A saddle-center for
α>4
Figure 1.3: Spectrum for the steady SH equation around u = ±1 for α > 1
For the uniform equilibrium u = 0 we have the same sequence, depending
again on the value of the parameter α: here the origin is a saddle-focus for
α < 0, a center for 0 < α < 1 and a saddle-center for α > 1. Note that for
α = 1, the two purely imaginary eigenvalues collapse at the origin.
1.2.2
Embedded solitons
Apart from uniform equilibria, equations (1.8) and (1.9) possess a rich structure of nonuniform equilibria. We postpone for now the description and the
discussion of the extensive literature developed to study this question.
In the literature we can find a considerable number of articles about kinks
connecting the equilibria u = −1 and u = 1 or pulses around these equilibria because the spectrum is far from the imaginary axis. In this way, one can
establish the existence of local stable and unstable manifolds around the equilibria. Therefore, the question about existence of kinks and pulses is equivalent
to the existence of intersections of these manifolds. These kind of results are
relatively strong and the local behavior around the equilibria is crucial, so that
they persist when varying the nonlinearities involved.
However, the question about homoclinic orbits to the equilibrium u = 0
is more delicate since it strongly depends on the nonlinearity involved and not
17
only on the local behavior around u = 0. In other words, different nonlinearities
with the same linearization can lead to different results.
While equilibria of equation (1.8) have been studied, equation (1.9) seems
more delicate and we will focus on it. Since this equation is still difficult to
handle, we will discuss a similar but simplified equation. We will state the
main results of the next chapter about existence and qualitative description of
homoclinic orbits converging to the origin, i.e., equilibria of (1.9) that verify
lim u(x) = 0.
x→±∞
(1.19)
Following the results of [17] and [46], we can think of these solitons as embedded
solitons in branches of homoclinic orbits to periodic solutions. Therefore, we
also discuss the existence of equilibria of (1.9) such that
lim u(x) − φ− (x) = 0,
lim u(x) − φ+ (x) = 0,
x→−∞
x→∞
(1.20)
where φ− (x) and φ+ (x) are periodic functions which can be determined a priori.
The concept of embedded solitons comes from the following fact: for some particular cases of those periodic solutions, they are reduced to the uniform state
u = 0, so that the solitons are embedded. In other words, the solitons are a
particular case of homoclinic orbits to periodic solutions.
Stimulated by the results from the literature and by the difficulty of proving
existence and describing the shape of the pulses for the EFK and SH equations,
we develop a new approach in the next Chapter. We keep our study for the
equation (1.14). But we substitute the original symmetric cubic nonlinearity
(1.11) by the symmetric piecewise nonlinear function

1


u+1
if
u6− ,


2


1
(1.21)
f (u) =
−u
if
|u| < ,

2



1

u−1
if
u> .
2
Like the cubic polynomial (1.11), this function is odd and has qualitatively the
same shape. The advantage of studying equation (1.14) with nonlinearity (1.21)
is that we can make explicit computations and obtain very precise results about
existence, multiplicity and geometric description of pulses.
Let us outline the procedure followed there. We search for even pulses that
cross the line u = 12 only once on the positive axis at, say, x = ζ > 0. By
18
symmetry we impose the conditions u0 (0) = u000 (0) = 0 and we put u(0) = α,
where α is still to be determined. We have the conservation of energy property
(equation (1.15)) but applied to a new potential function F derived from the
piecewise linear function (1.21). Thus, equation (1.6) can be explicitly written
as follows

1
1
1 2


u +u+
u6− ,


2
4
2

Z u

1 2
1
(1.22)
F (u) =
f (s) ds =
− u
|u| < ,

2
2
0



1

 1 u2 − u + 1
u> ,
2
4
2
which has been chosen so that F (0) = 0. Therefore, a pulse
p must verify E(u) =
E(0) = 0, so that u00 (0) is fixed and given by u00 (0) = ± 2F (α).
We use the fact that the orbit on x ∈ (0, ζ) and x ∈ (ζ, ∞) is described
by a linear differential equation that we can solve explicitly, including some
parameters. The next step is to fit all the parameters including α to obtain the
solution.
Let us recall the notation used in the next chapter. We study the SH equation
(1.9) via the rescaled version (1.18), with parameter q > 2 defined by (1.17). In
terms of the original equation (1.9), this corresponds to the parameter interval
α ∈ (1, 2). We consider the nonlinearity f given by (1.21). In our first theorem,
we state the existence of a sequence of homoclinic orbits to the equilibrium
u = 0.
Theorem 1.1 There exists an infinite sequence {qk > 2 : k = 1, 2, . . . } which
tends to infinity such that for q = qk equation (1.18), with f given by the
nonlinearity (1.21), has an even homoclinic orbit u(x) to the origin u = 0. It
has the following properties:
u>0
on
and
1
u(0) > 1 + √ ,
2
(1.23)
and
u0 (x) < 0
for
0 < x < ∞.
(1.24)
The plots of the first ten pulses are shown in Figure 1.4(a). In Figure 1.4(b)
we plot the second derivative of the orbits u00k .
As was previously mentioned, we can look at the pulses as embedded in
branches of homoclinic orbits to periodic solutions at x = ±∞. The periodic
19
Graphs of u(x)
u’’(k,x), k=1,3,5
0.2
1.8
1.6
–15
1.4
–10
–5
5
x
10
15
0
1.2
u 1
–0.2
0.8
u’’
0.6
–0.4
0.4
0.2
–20
–10
0
10
x
20
(b) Orbits for u00
k (x), k = 1, 3, 5
(a) 10 solitons
Figure 1.4: Homoclinic orbits to u = 0
orbits φ± (x) can be explicitly written as cosine functions. We define ε as the
amplitude of the periodic solution, ω(q) as the frequency of the periodic orbit,
and again ζ(q, ε) as the unique positive point at which the orbit crosses u = 12 ,
so u(ζ) = 12 . Then, we have the following result
+
Theorem 1.2 For every k > 1, there exists an interval [ε−
k , εk ] around ε = 0
− +
and an analytic function qk : [εk , εk ] → that satisfies qk (0) = qk such that for
+
every ε ∈ [ε−
k , εk ] there exists an even function uk (·, ε) that solves the differential
equation (1.18), with f given by (1.21), and q = qk (ε) and such that
uk (x, ε) = ε cos ω(q) x − ζ(q, ε) + o(1)
as
x → ±∞.
In Figure 1.5(a) we show one prototype of a homoclinic solution to a periodic
orbit. In Figure 1.5(b) we present the bifurcation diagram in which we plot q
versus the amplitude ε.
1.2.3
Analytical methods
Equation (1.14) has been studied in detail by many people. The first results
about equation (1.14) with f (u) = u − u2 , were established by a group of
20
PSfrag replacements
PSfrag replacements
ε
q
0.2
2
ε
0.1
1.5
3
4
q
5
6
0
u
1
–0.1
0.5
–0.2
–20
–10
0
10
x
20
(b) Bifurcation branches
(a) Homoclinic orbit to periodic solution
Figure 1.5: (a) A homoclinic orbit to a periodic orbit. (b) Bifurcation branches
of homoclinic orbits.
researchers from Bath, namely, Amick, Buffoni, Hofer and Toland in [4, 36, 11].
They study (1.18) with f given by the parabolic polynomial f (u) = u2 − u.
A second community of researchers, mainly coming from Leiden University,
namely J.B. van den Berg, L.A. Peletier, W.C. Troy, A.I. Rotariu-Bruma, J.
Hulshof, R.C.A.M. van den Vorst and others studied (1.14) with f given by the
cubic polynomial (1.11).
In [9], it is proved that in the parameter range β = 1 and γ ∈ (0, 81 ], the
set of bounded solutions allowed to exist are essentially the same as for the
classical FK equation, i.e. with γ = 0. This result is generalized for other
nonlinearity functions f similar to the cubic polynomial (1.11). Thus, we find
that for positive γ sufficiently small, the set of bounded orbits consists of the
three uniform equilibria, two monotone antisymmetric kinks and a family of
periodic orbits parametrised by the Energy E ∈ − 41 , 0 .
Note that the parameter range γ ∈ (0, 18 ] coincides precisely with the parameter range such that the equilibrium u = ±1 is a real saddle, as shown in
Figure 1.2(a).
Let us give a more geometric view of the results: the bounded solutions can
be represented by a picture similar to Figure 1.1(a): while general orbits can be
represented in the fourth dimensional space (u, u0 , u00 , u000 ) ∈ 4 , the bounded
21
orbits are uniquely projected on the second dimensional subspace (u, u0 ) ∈ 2 ,
like for the second order differential equation (1.2).
Partial results in this direction have also been obtained by singular perturbation theory [30, 41] and information about the stability of these orbits has
been obtained in [32, 1].
Existence and description of the shape and properties of solutions of the
canonical equation (1.14) has been done in a sequence of publications by L. A.
Peletier and W. Troy [55, 56, 57, 58] and collected in [59]. There the authors
make extensive use of the topological shooting method : this method is especially
appropriate for finding specific types of solutions with certain symmetries. To
search for odd solutions, they fix a priori u(0) = u00 (0) = 0 and put u0 (0) = α ∈
\ {0} and they fix the Energy E. By the conservation of the Energy, u000 (0) is
given by
αβ
4E + 1
u000 (0) =
−
,
2
4αγ
so that they can parametrise the orbits by their Energy. Let the parameters β
and γ be fixed and denote the solution of (1.14) by u(x, α). They study the
existence of values for α such that
lim u(x, α) = 1,
x→∞
to find kinks connecting the equilibria u = −1 and u = 1. To search for periodic
solutions, they study the existence of some value ζ(α) such that at x = ζ we
have u0 (ζ) = u000 (ζ) = 0.
To search for even solutions, they fix a priori u0 (0) = u000 (0) = 0 and u(0) =
α ∈ and the Energy E. By the conservation of the Energy, u00 (0) is given by
s (α2 − 1)2
2
00
E+
.
u (0) = ±
γ
4
In this case, the existence of values for α such that
lim u(x, α) = 1,
x→∞
leads to a homoclinic orbit to u = 1. By the method described above, they can
also study the existence of periodic solutions.
In all the previous cases, to search for homoclinic orbits to an equilibrium,
kinks and periodic solutions, the authors make use of the reversibility property:
equation (1.14) is invariant under the linear transformations
L1 : u(·) → −u(·),
L2 : u(·) → −u(− ·).
22
(1.25)
This invariance property allows us to extend the orbits mentioned above to all
x ∈ . This is one of the applications of finding certain types of symmetries in
the equation being considered. We will make larger use of this property along
the thesis.
On the other hand, imposing symmetries on the solution to be searched
(either an even or an odd solution) makes the search easier, since it decreases
the number of degrees of freedom. If we do not allow symmetries to be present,
then a two dimensional shooting method must be used.
When the authors study equation (1.14) in [55, 56, 57, 58, 51] they use the
concept of building block, that was introduced in [11]. This tool helps them to
classify the orbits in terms of the localization of the extrema and to describe
their shape. The analysis was originally developed in [11] for equation (1.14)
with nonlinearity
f (u) = u2 − u.
(1.26)
In [10] these techniques were applied for the equation (1.14) with nonlinearity
f given by the cubic polynomial (1.11).
In [36] the authors make use of degree theory and the antipodal mapping
theorem to study the existence of orbits, including pulses to an equilibrium. In
[52] the authors study this question by the shooting method and they prove
nonexistence of pulses for the EFK equation for β = 1 and γ ∈ 0, 81 , but they
prove the existence of pulses for nonlinearities of different type, including the
nonsymmetric cubic function
f (u) = (u2 − 1)(u + a),
(1.27)
with a ∈ (0, 1). In both cases, the pulses are homoclinic orbits to the equilibrium
u = ±1. Note that for this choice of parameters β and γ, the equilibrium u = ±1
is a real saddle, as shown in Figure 1.2(a). Thus, both the stable and unstable
manifolds are two-dimensional. On the other hand, linear analysis can give us
a good description of the asymptotic behavior of the pulses.
However, the question about existence of pulses to an equilibrium for different choices of parameters β and γ is more delicate and has been the object
of many studies in recent years from different perspectives. In this context,
we mention the work by Mielke, Holmes and O’Reilly [50] and the surveys of
Champneys [13] and [14] and the references quoted therein. Concerning equations such as (1.14), we mention a result of Amick & McLeod [3], and Hammersley & Mazzarino [33] (see also Eckhaus [24]), which proves the nonexistence
of such orbits for the function (1.26). We will also mention the work on the
Extended Fisher-Kolmogorov equation done in [43, 42].
23
For the cubic function f given by (1.11), Grotta-Ragazzo [61, 62] proved
that once a homoclinic orbit is found for some value β ∗ , there exist infinitely
many such orbits for values of β in any small neighborhood around β ∗ .
1.3
1.3.1
Pattern selection in the SH equation
Gradient systems
As was previously argued in Subsection 1.2.1 above, when studying PDE’s, we
are not only interested in the steady-state solutions but also in the dynamics.
Equilibrium solutions are particularly interesting when they are stable in some
sense (i.e., finding the appropriate space and topology) since they can be used
to determine the asymptotic behavior of the PDE. But often, the flow of the
PDE does not tend to an equilibrium, or it does under certain conditions, for
the appropriate set of initial conditions, etc.
There exist very few systems for which this property holds, and when it
does, it can be hard to prove it. But, as we mentioned before, one of those
systems for which this property holds, that often occurs in physical systems, is
a gradient system. The easiest example of a gradient system is a physical system
where we can define a functional operator J (usually called the Action), that is
decreasing (by some dissipation process) and for which the functional can not
decrease indefinitely but it has a lower bound.
In a gradient system we can define an operator J on some Hilbert space such
that the flow can be defined by the following equation
∂u(x, t)
= − ∇J u(x, t) ,
∂t
for x ∈ Ω,
t > 0.
(1.28)
Note that the function J is also a Lyapunov function.
It is well known that the Fisher-Kolmogorov and the Swift-Hohenberg equation are gradient systems. In this section we will consider a finite domain
Ω = (0, L), for some L > 0, and we study the Cauchy problem with initial
condition u(0, x) = u0 (x) for every x ∈ (0, L) and Dirichlet boundary conditions u(t, 0) = u(t, L) = 0 for every t > 0 for the FK equation. For the SH
equation we add
∂2u
∂2u
(t,
0)
=
u(t, L) = 0,
for every t > 0.
(1.29)
∂x2
∂x2
Note that solutions of this problem correspond to 2L-periodic solutions on the
real line, i.e. when we study the differential equation on x ∈ .
24
The SH equation can be read as a sum of a linear differential operator and
a reaction term −fα , where
fα (u) = (1 − α)u + u3 .
The action associated to the SH equation is given by
Z L
1 00 2
def 1
(u ) − (u0 )2 + Fα (u) dx
J(u, L) =
L 0
2
where
def
Fα (u) =
1.3.2
Z
u
0
−fα (s)ds =
1 − α 2 u4
u + .
2
4
(1.30)
(1.31)
(1.32)
The continuous model
The main motivation for this study has been the observations made in [53, 54].
The authors are interested in the asymptotic behavior of the dynamics for fixed
initial data and parameter α and varying parameter L. They restrict their
search for symmetric initial data, so u0 satisfies the following equation:
u(x) = u(L − x),
for every x ∈ (0, L).
(1.33)
By the form of the equation (1.9), the symmetry given by (1.33) holds for every
t > 0. Therefore, they are interested in symmetric equilibria.
The authors find a finite series of Gaps, i.e., a sequence of intervals of the
+
type [L−
k (α), Lk (α)] with k = 1, 2, . . . , n such that u(x, t) → 0 as t → ∞ if L ∈
−
+
[Lk (α), Lk (α)]. However, outside these Gaps, the limit behavior is a nontrivial
equilibrium. The purpose is to describe the selected pattern for different values
of L, the qualitative properties, and the local bifurcation diagram around the
extrema of the Gaps.
Besides the analytical approach, they are interested in a more global picture,
motivated by the numerical experiments. They study the map L → u∞ (·) L ,
where u∞ (·) represents the final pattern on (0, L). One of the relevant issues
observed is the existence of discontinuities in this map. More specifically, they
fix α and u0 and they draw the action or the norm of the final pattern versus
L, and they find that the plot is discontinuous at certain values of L.
Our intention in this section is to study more carefully a simplified model,
which is able to capture these phenomena and, at the same time, which is
sufficiently simple so that it can be carefully analyzed. In this way we try to
understand the relevant aspects of the “complex” continuous model by studying
the “simplified” discrete model in detail.
25
1.3.3
The discrete model
As mentioned above, our intention is to simplify equation (1.9). As an introduction, we will start with the simplification of the classical Fisher-Kolmogorov
equation (1.2), for which the dynamics is well known so that we can easily
compare the continuous and the discrete case. Then, we will follow a similar
procedure for the SH equation (1.9). In both equations, we consider the domain
Ω = (0, L).
For equation (1.2), we rescale the variables x = Lx∗ and t = L2 t∗ . This
transforms equation (1.2) into the following equation on the unit interval I =
(0, 1).
∂u
= ∆u + σf (u),
x∗ ∈ (0, 1),
t∗ > 0,
(1.34)
∂t∗
with σ = L2 . Here, the reaction term is given by f (u) = u − u3 . Note that we
have adopted a different convention than in (1.11). The operator ∆ represents
the Laplacian operator with respect to x∗ . From now on, we will drop the
asterisks when we refer to this equation.
We discretize equation (1.34) by defining a finite lattice of N + 1 nodes as
follows
def
un (t) = u(nh, t),
for n = 0, . . . , N + 1,
with
h=
1
.
N
(1.35)
We define the discrete Laplacian operator by
def
∆un = un+1 − 2un + un−1 .
(1.36)
Thus, the discrete Fisher-Kolmogorov system is defined as follows
dun (t)
= ∆un (t) + σf un (t) ,
dt
n = 1, . . . , N − 1.
(1.37)
For equation (1.9), we rescale space x = Lx∗ . This transforms equation (1.9)
into the following equation on the unit interval I = (0, 1).
∂u
= −γ 2 ∆2 u − 2γ∆u − fα (u),
∂t∗
x∗ ∈ (0, 1)
(1.38)
with γ = L12 and fα given by (1.30). Here, the operator ∆ represents the
Laplacian operator with respect to x∗ . From now on, we will drop the asterisks
when we refer to this equation.
26
We discretize equation (1.38) by defining again the finite lattice (1.35) of
N + 1 nodes. We use the discrete bi-Laplacian operator
def
∆2 un = ∆ ∆un = un+2 − 4un+1 + 6un − 4un−1 + un−2 ,
(1.39)
so that the discrete Swift-Hohenberg system is defined as follows
dun (t)
= −γ 2 ∆2 un (t) − 2γ∆un (t) − fα un (t) ,
dt
n = 1, . . . , N − 1, (1.40)
where we have defined the virtual nodes
u−1 (t) = −u1 (t) and uN +1 (t) = −uN −1 (t),
for every t > 0.
(1.41)
Note that we need to introduce these extra nodes for the system (1.40) to be
well defined. The choice given by (1.41) is the natural one for the boundary
conditions (1.29).
We will denote vectors in N −1 by u, and given a function f defined from
to , we will write
T
f (u) = f (u1 ), f (u2 ), . . . , f (uN −1 ) .
With this notation equation (1.37) can be rewritten as
u0 (t) = −Au(t) + σf (u),
(1.42)
where A = (aij ) is the symmetric positive definite matrix with ajj = 2 and
aj,j+1 = −1, aj+1,j = −1, and all other entries equal to 0. Its eigenvalues are
given by λj , where
πj
λj = 2 1 − cos
,
for j = 1, . . . , N − 1.
N
We can define the action J of the discrete Fisher-Kolmogorov system (1.37)
and the discrete Swift-Hohenberg system (1.40). First, we introduce the operators J1 and J2
def
J1 (u) =
N
1 T
1 X
(un − un−1 )2 ,
u Au =
2
2 n=1
def
J2 (u) =
N −1
1 T 2
1 X
(∆un )2 .
u A u=
2
2 n=1
27
(1.43)
(1.44)
Then, the action associated to (1.37) is given by
def
JF (u) = J1 (u) − σ
N
−1
X
F (un ),
(1.45)
n=1
where the potential F is defined by (1.6). Similarly, the action associated to
(1.40) is given by
def
JS (u) = γ 2 J2 (u) − 2γJ1 (u) +
N
−1
X
Fα (un ),
(1.46)
n=1
where the potential Fα is defined by (1.32). The discrete FK system (1.37) is
a Gradient system for every σ > 0 while the discrete SH system (1.40) is a
Gradient system for every α ∈ and γ ∈ . Therefore, this property holds for
both the discrete and continuous equations.
in
An extra property holds for both cases. Define the following linear operators
N −1
.
L1 (u1 , u2 , . . . , uN −2 , uN −1 ) = (−u1 , −u2 , . . . , −uN −2 , −uN −1 ),
L2 (u1 , u2 , . . . , uN −2 , uN −1 ) = ( uN −1 , uN −2 , . . . , u2 , u1 ),
L3 (u1 , u2 , . . . , uN −2 , uN −1 ) = (−uN −1 , −uN −2 , . . . , −u2 , −u1 ).
(1.47)
Note that L3 = L1 L2 = L2 L1 . The unique fixed point of L1 is the origin, while
the fixed points of the other operators are given by
Π2
def
Π3
def
=
=
u∈
u∈
N −1
N −1
: uj = uN −j
: uj = −uN −j
for every j = 1, . . . , N − 1 ,
for every j = 1, . . . , N − 1 .
(1.48)
Invariance of the flow with respect to Π2 in the discrete model corresponds to the
property (1.33) for every t > 0 in the continuous model. Similarly, invariance of
the flow with respect to Π3 in the discrete model corresponds to the following
property holding for every t > 0 in the continuous model
u(x) = −u(L − x),
for every x ∈ (0, L).
(1.49)
The elements of Π2 will be called symmetric and the elements of Π3 will be
called antisymmetric.
From now on, we study in detail the case N = 4. In this case, the symmetric
equilibria can be represented as points in 2 . Let us summarize the main results
28
about equilibria of the discrete FK equation (1.37) in the following theorem. In
Figure 1.6 we draw the bifurcation diagram of the symmetric equilibria. There
we represent the norm of the equilibria ku(σ)k versus the parameter σ. In
dashed line we draw the unstable equilibria and in solid line the stable ones.
Theorem 1.3 Let N = 4. The symmetric equilibria of equation (1.37) have
the following properties:
√ 1. For σ ∈ 0, 2 − 2 , the origin is the unique symmetric equilibrium. For
√
σ > 2 − 2, the origin is unstable and a pair of nontrivial symmetric
equilibria ±u∗ (σ) bifurcate from the origin. They are located in the first
and third quadrant and they are the√global symmetric mimimisers of J F
defined by (1.45) for every σ > 2 − 2.
√
2. For σ > 2 + 2, a second pair of nontrivial symmetric equilibria ±ζ(σ)
bifurcate from the origin. They are saddles located in the second and fourth
quadrant.
√
3. There exists σ̃ > 2 + 2 such that for σ > σ̃, two pairs of nontrivial symmetric equilibria emerge from a saddle-node bifurcation. They are located
in the second and fourth quadrant. Therefore, there exist symmetric initial
data for which the flow converges to an equilibrium that has not one sign.
Property 1 mirrors the continuous FK model, while Property 3 is qualitatively
specific for the discrete FK model.
Our main interest is focused on the discrete SH system (1.40). Like in [53, 54], for
fixed α and every N we are able to find a sequence of Gaps such that u(t) → 0 if
the parameter γ lies in one of those Gaps and it tends to a nontrivial equilibrium
otherwise.
We find that for given N , for α > 0 sufficiently small, there exists at least one
Gap. This case is relatively simple, since all the nontrivial equilibria bifurcate
from the origin and the final state φ∞ (u0 , γ) is continuous with respect to γ.
Moreover, the values of γ for which the bifurcation occurs is determined by
the linearized operator around the origin and the shape is determined by the
corresponding eigenvector. But when pushing α higher, Gaps start to dissapear
and new phenomena occur. It is rather difficult to describe the process for high
N and we will consider N = 4.
We introduce the following critical values of γ
√ √
def
γ1± (α) =
1 + 22 (1 ± α) .
(1.50)
√ √
def
γ3± (α) =
1 − 22 (1 ± α) .
29
2
||u||
1.5
1
0.5
PSfrag replacements
0
0
5
10
15
20
σ
Figure 1.6: Bifurcation diagram of the symmetric equilibria
For every N , there exist N − 1 pairs of critical values: the odd ones correspond
to symmetric eigenvectors and the even critical values with antisymmetric eigen±
±
−
+
vectors. For
this reason, we consider γ1 and γ3 . Note that γ1 (α) > γ3 (α) iff
1
α ∈ 0, 2 . Therefore, we can distinguish two parameter ranges for α: 0 < α < 21
and 21 < α < 1. In Figure 1.7, the critical values correspond to the bifurcation
points of the diagram and they are ordered as follows: γ3− < γ3+ < γ1− < γ1+ .
We start by considering the interval 0 < α < 21 . Take γ > γ1+ . Then,
the origin is the unique equilibrium. However, when γ drops below γ1+ two
nontrivial solutions ±u1 (γ) bifurcate from the origin, one into the first quadrant,
and one into the third. They continue to exist until γ reaches γ1− , when they
merge with the trivial solution again. For γ ∈ [γ3+ , γ1− ] the origin remains the
only equilibrium state, but at γ3+ two nontrivial solutions u3 (γ) bifurcate again
from the origin, this time into the second and the fourth quadrant. They too,
return to the trivial solution, at γ3− . For γ below γ3− the trivial solution is the
unique equilibrium state. The situation is illustrated in a bifurcation diagram
in Figure 1.7.
Theorem 1.4 Let α ∈ 0, 12 and γ > 0.
(a) If γ ∈
/ γ1− , γ1+ ∪ γ3− , γ3+ , then the origin is the unique equilibrium state,
30
1
0.9
0.8
0.7
kuk
0.6
0.5
0.4
0.3
PSfrag replacements
0.2
0.1
0
0
0.5
1
γ
1.5
2
2.5
3
Figure 1.7: Graph of kuk versus γ for α = 0.3
and it atracts all orbits.
(b) If γ ∈ γ1− , γ1+ , then there exist two equilibrium states ±u1 (γ), one in the
first quadrant, and one in the third. The origin is now unstable and ±u1
are stable.
(c) If γ ∈ γ3− , γ3+ , then there exist two equilibrium states ±u3 (γ), one in the
second quadrant, and one in the fourth. The origin is unstable and ±u3 are
also unstable.
Next, let us study the interval 21 < α < 1. Again, the origin is the unique
equilibrium for γ > γ1+ and when γ drops below γ1+ two nontrivial solutions
±u1 (γ) bifurcate from the origin. But then, at γ3+ two nontrivial solutions
u3 (γ) bifurcate again from the origin and before they both return to the trivial
PSfrag replacements
solution, twopairs of new solutions emerge by a saddle-node bifurcation at some
γ∗+ ∈ 12 , γ3+ . For 0 < γ < 21 the bifurcation phenomena is reversed and the
eight nontrivial solutions gradually dissapear. This is described in the following
theorem. For convenience, we put the critical values of γ on a γ-axis. Note
(a)
0
(b)
γ3−
(c)
γ1−
(d)
γ∗−
(d)
1/2
(c)
γ∗+
Figure 1.8: Critical values of γ.
31
(b)
γ3+
(a)
γ1+
that for γ = 1/2, the system of ODE’s consists of two independent differential
equations.
Theorem 1.5 Let α ∈ 21 , 1 and γ > 0. Then, we identify 6 critical values of
γ:
0 < γ3− (α) < γ1− (α) < γ∗− (α) < γ∗+ (α) < γ3+ (α) < γ1+ (α),
such that:
(a) If γ ∈
/ γ3− , γ1+ , the origin is the unique equilibrium state and it attracts all
orbits.
(b) If γ∗− < γ < γ1+ , a pair of nontrivial equilibria ±u exist, one in the first
quadrant and one in the third. At γ = γ∗− , u has a saddle-node bifurcation
and at γ1+ , it has a supercritical bifurcation from the origin. u is a stable
equilibrium and it is the minimizer for 21 < γ < γ1+ .
(c) If γ1− < γ < γ3+ , a pair of nontrivial equilibria ±y exist. For γ1− < γ < 12 ,
they are located in the second quadrant (respectively the fourth) and for
+
1
2 < γ < γ3 , they are located in the first quadrant (respectively, the third).
They are all saddles.
(d) If γ3− < γ < γ∗+ , a pair of nontrivial equilibria ±z∗ exist, one in the second
quadrant and one in the fourth. At γ = γ∗+ , it has a saddle-node bifurcation
and at γ = γ3− it has a supercritical bifurcation from the origin. The points
±z∗ are stable and they are minimizers for γ3− < γ < 21 .
(e) If γ∗− < γ < γ∗+ , a pair of nontrivial equilibria ±z∗∗ exist. For γ1− < γ < 12 ,
they are located in the second quadrant (respectively the fourth) and for
+
1
2 < γ < γ3 , they are located in the third quadrant (respectively, the first).
They are saddles.
A simple bifurcation theorem shows that around the critical values γ1− , γ1+ ,
γ3+ there exist local bifurcation branches. These branches can be continued
as long as the conditions for the Implicit Function Theorem hold, that is, as
long as the appropriate determinant does not vanish. Applying this argument
to this case, we find by a combination of analytical and numerical arguments
that one branch can be extended smoothly along the interval [γ1− , γ3+ ], and the
other one can be extended continuousy but not smoothly along [γ3− , γ1+ ].
This is summarised in the following proposition:
γ3− ,
32
1.5
||u||
1
0.5
PSfrag replacements
0
0
0.2
0.4
0.6
0.8
1
γ
Figure 1.9: Graph of kuk versus γ for α = 0.7
Proposition 1.1 There exists an smooth parametrization h1 : [0, 1] → [γ1− , γ3+ ]×
3
with components s → (γ, u1 , u2 , u3 ) and a continuous (but not smooth)
parametrization h2 : [0, 1] → [γ3− , γ1+ ] × 3 such that
(a) h1 (0) = (γ1− , 0, 0, 0),
positive on s ∈ (0, 1).
h1 (1) = (γ3+ , 0, 0, 0), and
(b) h2 (0) = (γ3− , 0, 0, 0),
s ∈ (0, 1).
h2 (1) = (γ1+ , 0, 0, 0), and
∂γ
∂s
∂γ
∂s
is continuous and
is discontinuous on
(c) h1 (·) and h2 (·) represent all the symmetric equilibria of the discrete SH
equation, up to symmetry. More specifically, let (γ, u) be a nontrivial symmetric equilibrium of the discrete SH equation. Then, there exists a unique
s ∈ (0, 1) and a unique i ∈ {1, 2} such that either hi (s) = (γ, u) or
hi (s) = (γ, −u).
In Figure 1.9, we can see the bifurcation diagram of the norm of all the equilibria.
Here, we show ku(γ)k versus γ for α = 0.7. The equilibria bifurcating from the
origin correspond to γ3− < γ1− < γ1+ . The values γ∗− < γ∗+ are those for which
the branch of equilibria changes stability.
33
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−1
0
1
2
x
3
−0.5
−1
0
4
(a) γ = 4
1
2
x
3
−1
0
4
(b) γ = 3
1
0.5
0.5
0.5
0
0
0
−1
0
u
1
−0.5
−0.5
1
2
x
(d) γ = 0.47
3
4
−1
0
1
2
x
3
4
(c) γ = 0.48
1
u
u
u
1
u
u
1
−0.5
1
2
x
(e) γ = 0.05
3
4
−1
0
1
2
x
3
4
(f) γ = 0.01
Figure 1.10: Dynamics for α = 0.7 and different values of γ.
Let us finish this section with some remarks about the final profile. Let the
initial value u0 ∈ 3 , u0 6= (0, 0, 0) be fixed and symmetric, and consider the
map
γ → φ∞ (u0 , γ),
for every γ > 0.
(1.51)
As we have mentioned above, in the continuous model there were observed
discontinuities with respect to γ. Now we are able to describe
the reason for this
for the discrete model. Let us restrict to the case α ∈ 21 , 1 . Then, as described
in Theorem 1.5, there exist four different stable symmetric equilibria ±u and
±z∗ on γ ∈ (γ∗− , γ∗+ ). Moreover, φ∞ (u0 , γ1− ) = ±u and φ∞ (u0 , γ3+ ) = ±z∗
for almost every initial u0 , since they are the only stable equilibria. Therefore,
there exists some γ ∈ (γ∗− , γ∗+ ) where the final profile switches and the map
(1.51) has at least one discontinuity.
We show this evolution of the final profile in Figure
1.10. We fix the ini
tial condition u(0) = sin π4 , sin π2 , sin π4
= √12 , 1, √12 and α = 0.7
and we plot φ∞ (u0 , γ) for different values of γ in thick line. As γ decreases,
34
φ∞ (u0 , γ) bifurcates from the origin and it has a discontinuity at some value
γ ∈ (0.47, 0.48). As γ is reduced further, φ∞ (u0 , γ) becomes the origin.
This problem is a two-dimensional system of ODEs and we can treat it
with phase-plane analysis. We use a combination of numerical and analytical
arguments. For every γ ∈ (γ3− , γ1+ ), apart from the origin, all the equilibria
are stable nodes and saddles. Let us call Ui the domains of attraction of the
stable nodes and Ws (Pi ) the stable manifolds of a saddle Pi . Then, we have the
following proposition
Proposition 1.2 Let γ0 > 0, γ0 6= γ∗− , γ∗+ and u0 ∈ 3 , u0 =
6 (0, 0, 0) be
symmetric. Then
the
map
(1.51)
is
discontinuous
at
γ
=
γ
if
and only if
0
u0 ∈ Ws Pi (γ0 ) for some saddle Pi (γ0 ).
1
1
0.5
0.5
u2
u2
This proposition is first shown for a noncritical value γ0 and the main ingredient
is that the domain of attraction of a node is open. The result can be extended
to γ3− , γ3+ , γ1− , γ1+ because the equilibria bifurcate continuously from the origin.
As for γ = γ∗− , γ∗+ , we have a saddle-node bifurcation, this case is qualitatively
different.
0
−0.5
−1
−1
0
−0.5
−0.5
0
u1
0.5
−1
−1
1
(a) γ = 0.475.
−0.5
0
u1
0.5
1
(b) γ = 0.47.
Figure 1.11: Graphs of the orbits for the discrete SH equation with α = 0.7.
The case γ = γ∗− is best treated by analyzing the behaviour of the boundaries
of Ui (γ) for γ > γ∗− but close to γ∗− . We have that the map (1.51) is continuous
on [γ∗− , γ∗− +ε) for some ε > 0. Consider the unique stable node Pi (γ) contained
in the i-th quarter and Ui its corresponding domain of attraction. Take u0 ∈
35
U1 (γ∗− ). Then we have that there exists some ε > 0 such that
φ∞ (u0 , γ) = P1
φ∞ (u0 , γ) = P4
for
for
γ ∈ [γ∗− , γ∗− + ε),
γ ∈ (γ∗− − ε, γ∗− ).
This is observed in Figure 1.11. There, the isoclines are represented by dashed
lines and the orbits by thick lines.
1.4
1.4.1
Travelling waves on a lattice
Lattices
We recall that the purpose of this thesis is to study some generalizations of
the second order partial differential equation (1.2) that are used to described
different physical phenomena. Previously we have worked with fourth order differential equations. In this section we want to study a different type: differential
equations on a lattice. When the lattice is finite, we have a system of a finite
number of differential equations. When the lattice is infinite, we have an infinite
number of them. We want to study this case in detail. As before, we do not
attempt to cover all the possible cases but we restrict our study to a particular
one.
Note that we can also view the previous sections as studies of differential
equations on a lattice, where the lattice is the open interval Ω for the continuous
models (1.8) and (1.9) and the lattice is {1, 2, . . . , N − 1} for the discrete models
(1.37) and (1.40). In this section we will consider the discrete lattice .
1.4.2
Fronts on a lattice
We consider the following equation
0
un (t) = −un (t) + un−1 (t) + f un−1 (t) ,
un (0) =
an ∈ [0, 1]
t > 0, n ∈
n∈
,
,
(1.52)
where f is given by (1.26).
Problem (1.52) arises in the context of a clock model for a dilute gas of N
particles with short range interactions, in which every particle carries a clock
with a discrete time n ∈ which is advanced at every collision. This happens
according to the following rule: when two particles collide, they both reset their
respective clock values, say n and `, to either n + 1 or ` + 1, whichever is the
36
largest. Thus, if we denote the number of particles with clock value n by Nn ,
we obtain the following dynamical equation:
∞
n−1
X
X
dNn
=−
Rn,` − 2Rn,n + 2
Rn−1,` ,
dt
`=−∞
`6=n
`=−∞
where Rn,` denotes the rate by which collisions occur between particles with
clock values n and `. We assume this rate to be proportional to Nn N` /N 2 when
n 6= ` and to Nn2 /(2N 2 ) when two particles with equal clock value n collide.
Then, writing fn = Nn /N and scaling the time appropriately, we are led to the
equation
dfn
2
= −fn + fn−1
+ 2fn−1 un−2 = −fn + u2n−1 − u2n−2 ,
dt
where we have set
n
X
un =
f` .
`=−∞
Adding the equations for f` for all values of ` 6 n then yields
dun
= −un + u2n−1 ,
dt
(1.53)
which is equivalent to (1.52). For further details of this model we refer to [70].
For our analysis, in (1.52), we replace the nonlinear parabolic function (1.26)
by the following piecewise linear function
(
−s
for s 6 21 ,
f (s) =
(1.54)
s − 1 for s > 12 .
By definition of un , we are interested in increasing solutions with respect to
n, i.e.,
(1.55)
un (t) 6 un+1 (t),
for every n ∈ , t > 0,
with upper and lower bounds
0 6 un (t) 6 1,
for every n ∈
,
t > 0,
(1.56)
and the following assymptotic behaviour
lim un (t) = 0
for every t > 0,
(1.57)
lim un (t) = 1,
for every t > 0.
(1.58)
n→−∞
n→∞
37
For our first results, we impose conditions (1.55), (1.56), (1.57), (1.58). Then,
we will give some results for nonmonotone solutions, so we drop (1.55). Finally,
we will also study positive solutions that satisfy (1.57).
The uniform equilibria of equation (1.52) with nonlinearities given by (1.26)
and (1.54) are un = 0 and un = 1 for every n ∈ . We are interested in
searching for travelling waves connecting both equilibria, i.e., solutions that can
be written as
ϕ(ξ) = un (t),
ξ = n − ct,
(1.59)
for some c ∈ . The parameter c is called the wave speed.
We define the critical wave speed c0 as the unique positive root of the equation
c e1/c = 2 e
(c0 = 4.31107 . . . ).
(1.60)
We establish the first result
Theorem 1.6 For each c > c0 equation (1.52), with f given by (1.54) has a
unique monotone travelling wave un (t) = ϕ(ξ) of the form given in (1.59), such
that ϕ(−∞) = 0 and ϕ(∞) = 1.
Our next objective is to describe the existence and qualitative properties of
the waves for c < c0 . Since the dominant eigenvalues are not real, we can expect
waves with oscillatory tails around u = 1 as x → ∞. The existence is based
on a continuity argument and Theorem 1.6. Thus, we can prove the following
result
Theorem 1.7 There exists some c∗ ∈ (0, c0 ) such that for each c∗ < c < c0
equation (1.52), with f given by (1.54) has a unique nonmonotone travelling
wave un (t) = ϕ(ξ) of the form given in (1.59), such that ϕ(−∞) = 0 and
ϕ(∞) = 1.
We conjecture that c∗ equals cbif , the highest value for which the dominant
eigenvalues reach the imaginary axis.
To study the existence of waves such that ϕ(−∞) = 0 but the extra boundary condition (1.58) is dropped. Here, we mix analytical and numerical arguments: the analytical arguments are based on some topological condition and
use numerics to find the values c for which the condition holds.
We introduce the following critical wave speeds
0 < cunb < cper < cbif < c0 .
38
PSfrag replacements
cunb
2
c=1.4
c=c_unb
c=1.5
c=c_per
c=c_bif
c=2
c=5
1.5
1
0.5
0
-0.5
-4
-2
0
2
4
6
8
10
12
14
Figure 1.12: Fronts for different values of c
The numerical values are found to be
c0 = 4.311,
cbif = 1.654,
cunb = 1.455.
We have the following result
Theorem 1.8 We have
1. For each 0 < c < cunb , there exists a travelling wave un (t) = ϕ(ξ) of the
form given in (1.59), such that ϕ(−∞) = 0 and ϕ(∞) = −∞.
2. For c = cunb , there exists a travelling wave un (t) = ϕ(ξ) of the form given
in (1.59), such that ϕ(−∞) = 0 and x̃ such that u(x, cunb ) = 1 for every
x > x̃.
3. For each cunb < c < cper , there exists a travelling wave un (t) = ϕ(ξ) of
the form given in (1.59), such that ϕ(−∞) = 0 and there exist T (c) > 0
and x̃(c) ∈ such that ϕ(x + T (c), c) = ϕ(x, c) for every x > x̃(c).
39
4. For each cper < c < cbif , there exists a travelling wave un (t) = ϕ(ξ) of the
form given in (1.59), such that ϕ(−∞) = 0 and ϕ(·, c) > 0.
In Figure 1.12 we show the plots of the existing fronts for different values of
the wave speed c.
40
Chapter 2
Embedded solitons
2.1
Introduction
In recent years there has been a considerable interest in fourth order model
equations such as the Extended Fisher-Kolmogorov equation
∂u
∂4u ∂2u
= −γ 4 + 2 + u − u3 ,
∂t
∂x
∂x
γ > 0,
(2.1)
proposed in 1988 by Dee & Van Saarloos [22] as a model for bi-stable systems,
and the Swift-Hohenberg equation
∂u
∂ 2
= κu − 1 + 2 u − u3 ,
∂t
∂x
κ∈
,
(2.2)
proposed by Swift & Hohenberg [66] in studies of hydrodynamic instability.
Both equations are used in the description and analysis of complex patterns
[20, 25, 21, 31, 60] kinks, or heteroclinic orbits, which connect two different
uniform states, pulses, or homoclinic orbits to one of the uniform states, and
periodic patterns with a nontrivial structure. For an overview of recent results,
especially about stationary patterns, we refer to [9] and [59] and the literature
cited therein.
After an appropriate transformation, stationary solutions u(x) of (2.1) and
(2.2) are found to be solutions of the equation
uiv + qu00 + f (u) = 0,
41
q∈
,
(2.3)
in which primes denote differentiation with respect to x. For the ExtendedFisher Kolmogorov equation the transformation involved is
x = γ 1/4 x∗ ,
1
u(x) = u∗ (x∗ ) and q = − √ ,
γ
(2.4)
and for the Swift-Hohenberg equation, when κ > 1, the scaling is
x = (κ − 1)−1/4 x∗ ,
u(x) =
√
κ − 1 u∗ (x∗ ),
q=√
2
.
κ−1
(2.5)
In both cases the function f (s) becomes the cubic
f (s) = s3 − s.
(2.6)
In the present paper we analyse the existence of homoclinic orbits of equation
(2.3) which tend to the origin u = 0 as x → ±∞ for a specific form of f given in
(2.7) below. Homoclinic
to the uniform states u = ±1 have been obtained
√ orbits
√
for values of q ∈ (− 8, 8) in [59] and [63]. For these values of q the states
u = 1 and u = −1 are saddle-foci, with two-dimensional stable and unstable
manifolds. This case has been studied in detail in [42]. In contrast, u = 0 is a
saddle-center for all values of q and the stable and unstable manifolds are both
one-dimensional. Thus, the existence of homoclinic orbits to u = 0 is less likely.
We shall have to carefully tune the parameter q in (2.3) in order for a pulse to
exist.
Homoclinic orbits to saddle-centers have attracted a great deal of attention.
We mention the paper by Mielke, Holmes and O’Reilly [50] and the surveys of
Champneys [13] and [14] and the references quoted therein. Concerning equations such as (2.3), we mention a result of Amick & McLeod [3], and Hammersley
& Mazzarino [33] (see also Eckhaus [24]), which proves the nonexistence of such
orbits for the function
f (s) = s2 − s.
For the cubic function f of (2.6) Grotta-Ragazzo [62, 61] proved that once
a homoclinic orbit is found for some value q ∗ there exist infinitely many such
orbits for values of q in any small neighbourhood around q ∗ .
For a related equation
1 iv
3
u + u00 − λ2 u + u3 + u(uu00 − u02 ) = 0
12
4
Kivshar et. al. [45] established numerically the existence of homoclinic orbits
for the equation for discrete values of λ, and for a similar equation, arising in
42
the theory of water waves, with a second order nonlinear term, we mention the
work of Champneys & Groves [16].
In the present paper we study equation (2.3) when f is given by the piecewise linear function

1


s6− ,
s+1

2


1
(2.7)
f (s) =
−s
|s| < ,

2



1

s−1
s> .
2
Like the cubic function (2.6), this function is odd and has qualitatively the same
shape. However, since it is piecewise linear, we can make explicit computations,
and so obtain very precise results about existence and multiplicity of pulse-type
solutions.
Since we seek homoclinic orbits to the constant solution u = 0, the local
behaviour around u = 0 is important. Linearising around the origin we obtain
the equation
uiv + qu00 − u = 0,
with the associated characteristic equation
λ4 + qλ2 − 1 = 0.
It has two real roots λ = ±µ, and two imaginary roots λ = ±iω. Here
q
q
p
p
1
1
µ(q) = √
−q + q 2 + 4
and
ω(q) = √
q + q 2 + 4.
2
2
(2.8)
For this reason, u = 0 is called a saddle-center. Thus, the stable and the
unstable manifold are both one-dimensional, and it is not evident that they will
intersect to form a homoclinic orbit. We shall find that this happens at an
infinite sequence {qk } of values of q, which tend to infinity as k → ∞.
Specifically we prove the following theorem:
Theorem 2.1 There exists an infinite sequence {qk > 2 : k = 1, 2, . . . } which
tends to infinity such that for q = qk equation (2.3), with f given by the nonlinearity (2.7), has an even homoclinic orbit u(x) to the origin u = 0. It has the
following properties:
u>0
on
and
43
1
u(0) > 1 + √ ,
2
(2.9)
u0 (x) < 0
for
PSfrag replacements
PSfrag replacements
and
0 < x < ∞.
(2.10)
The graphs of the first ten homoclinic orbits are shown in Figure 2.1(a).
0.2
1.8
1.6
–15
1.4
–10
–5
5
x
10
15
0
1.2
u 1
–0.2
0.8
u’’
0.6
–0.4
0.4
0.2
–20
–10
0
10
x
20
(b) Orbits for u00
k (x), k = 1, 3, 5
(a) 10 solitons
Figure 2.1: Homoclinic orbits
Whilst for every k > 1, uk and u0k have one sign on + , the second derivative
exhibits more and more oscillations as k increases. In Figure 2.1(b), we show
for k = 1, 3, 5. In Appendix C we present a table of values of qk , ζk and of
uk and u00k at the origin.
u00k
u00k
The asymptotic behaviour of the critical values qk as k → ∞ is given by
qk =
4
2
k + + O(k −1 )
3
3
as k → ∞.
(2.11)
In view of (2.9) and (2.10) the graphs of the solutions uk (x) each intersect
the level line u = 21 precisely once (for x > 0), at a point which we shall denote
by x = ζk = ζ(qk ), i.e.
uk (ζk ) =
1
2
for k = 1, 2, 3, . . . .
(2.12)
We find that ζk → ∞ as k → ∞, with the following asymptotic behaviour:
ζ(qk ) =
3π √
−3/2
qk + O(qk
)
4
44
as k → ∞,
(2.13)
or
√
π 3√
ζk =
k + o(1)
as k → ∞.
2
On the other hand, as k increases, the maxima uk (0) decrease, and
1
uk (0) & 1 + √
2
as k → ∞.
(2.14)
(2.15)
Thus, the graphs of uk (x) become flatter and wider as k → ∞. Scaling the
independent variable x according to
x
y=√
q
and
vk (y) = uk (x),
we find that
vk (y) → V (y)
as
k→∞
uniformly in C 1 ( ), where V is the unique even positive solution of the problem
V 00 (y) + f (V (y)) = 0,
V (±∞) = 0.
As discussed in [46] and [17] (see also [13] and [14]) homoclinic orbits —
or solitons — can be viewed as embedded in branches of homoclinic orbits to
periodic orbits at x = ±∞. One such homoclinic orbit is shown in Figure 2.2(a).
In Section 5 we establish the existence of branches of such homoclinic orbits.
They are shown in Figure 2.2(b), where the vertical axis gives the amplitude ε
of the limiting periodic solution.
An interesting question concerns the phase shift 2φ between the limiting
periodic solutions at x = −∞ and x = +∞. In Figure 2.3 we show 2φ versus
the parameter q along the first four branches. The diamonds on the branches
correspond to the solitons.
Acknowledgement. The authors are grateful to Alan Champneys of Bristol
University for a number of stimulating discussions.
2.2
Preliminaries
We seek even homoclinic orbits to the origin, i.e. solutions u(x) of equation
(2.3) endowed with the symmetry property
u(−x) = u(x)
for x ∈
45
.
(2.16)
0.2
2
ε
1.5
u
PSfrag replacements
ε = 0.1
PSfrag replacements
ε
q
ε = 0.1
0.1
3
4
q
5
6
0
1
–0.1
0.5
–0.2
–20
–10
0
10
x
20
(a) Homoclinic orbit to periodic solution
(b) Bifurcation branches
Figure 2.2: (a) A homoclinic orbit to a periodic orbit. (b) Bifurcation branches
of homoclinic orbits.
5
4
3
2
1
0
3
4
q
5
6
–1
Figure 2.3: The phase shift 2φ versus q along the first four branches
Thanks to the invariance of the equation with respect to the transformation
u → −u, we may assume without loss of generality that u is positive at the
origin. In addition we assume that u has a local maximum at the origin. Thus
u(0) > 0
and
u00 (0) < 0.
(2.17)
As a first observation, we derive a lower bound for any critical value of such
a solution. The essential ingredient here is the first integral of equation (2.3),
46
which we shall often refer to as the Energy Identity. It is given by
1
q
def
E(u) = u0 (x)u000 (x) − {u00 (x)}2 + {u0 (x)}2 + F (u(x)) = E,
2
2
in which E is a constant, and F is the primitive of f , given by

1
1
1 2


u +u+
u6− ,


2
4
2

Z u

1
1
F (u) =
f (s) ds =
|u| < ,
− u2

2
2
0



1
1
1

2
 u −u+
u> ,
2
4
2
(2.18)
(2.19)
which has been chosen so that F (0) = 0. Thus, a homoclinic orbit u(x) which
converges to the origin as x → ±∞ has zero energy, i.e. E = 0.
Lemma 2.1 Let u(x) be a homoclinic orbit to u = 0, and let u0 (a) = 0 for
some a ∈ . Then either u(a) = 0 or
1
|u(a)| > 1 + √ .
2
(2.20)
Proof. We evaluate the energy at x = a, and remember that u0 (a) = 0. This
yields
1
− {u00 (a)}2 + F (u(a)) = 0,
(2.21)
2
so that
1
F (u(a)) = {u00 (a)}2 > 0.
(2.22)
2
From the definition of F we see that F (s) < 0 for s ∈ (−s0 , s0 ) \ {0}, where
s0 = 1 + √12 . We therefore conclude that either u(a) = 0 or |u(a)| > s0 , as
asserted.
An immediate consequence of Lemma 2.1 is that homoclinic orbits to u = 0
have monotone tails. In particular, we have
Lemma 2.2 Let u(x) be a nonnegative homoclinic orbit.
points x1 and x2 (x1 < x2 ) such that
u0 > 0
and
for
x < x1
1
u(x) > 1 + √
2
and
for
47
u0 < 0
for
x1 < x < x2 .
Then there exist
x > x2 ,
Plainly, if u(x) is even, then x1 < 0 and x2 > 0 and x2 = −x1 .
In the present paper we shall always assume that q > 2, and the question
remains as to the existence of homoclinic orbits when q 6 2, and the uniform
states u = ±1 are no longer saddle-centers, but saddle-foci. For the cubic
function (2.6), one can establish a lower bound for q [43].
Lemma 2.3 Let u(x) be a nontrivial homoclinic orbit to the origin that solves
the differential equation (2.3) in which the nonlinearity f is given by the cubic
polynomial in (2.6). Then
√
2 7
.
(2.23)
q>
3
For completeness we give the proof in Appendix A.
2.3
Existence of homoclinic orbits
In this section we turn to the question of the existence of homoclinic orbits of
equation (2.3) which tend to the origin as x → ±∞. We consider even solutions,
and hence it will suffice to find solutions of the following Initial Value Problem:
 iv
00
for
x > 0,

 u + qu + f (u) = 0
u(0) = α, u0 (0) = 0, u00 (0) = β, u000 (0) = 0,
(2.24)


u(x) → 0
as x → ∞,
where α and β are so chosen that the Energy Identity (2.18) is satisfied at the
origin, i.e.
1
− β 2 + F (α) = 0.
(2.25)
2
We assume that u(x) attains its maximum at the origin, and hence, that β =
u00 (0) < 0. We therefore conclude from (2.24) that F (α) > 0, and hence that
1
α>1+ √
2
and
β=−
p
2F (α).
(2.26)
We establish the existence of nonnegative solutions u(x) of Problem (2.24)
by explicitly constructing them. By Lemma 2.2 the graph of u(x) intersects the
48
line u = 12 only once on + , and as before (see (2.12)), we denote this point of
intersection by ζ. We then put
(
u+ (x)
for
0 6 x 6 ζ,
u(x) =
(2.27)
u− (x)
for
ζ 6 x < ∞.
By the definition of f (s) in (2.7), the function u− solves the equation
uiv + qu00 − u = 0.
Thus, since u− (ζ) =
1
2
it follows that
u− (x) =
1 −µ(x−ζ)
e
2
for
ζ < x < ∞,
(2.28)
where µ has been defined in (2.8).
We write u+ (x) = 1 + v(x). Then the function v(x) solves the equation
v iv + qv 00 + v = 0.
Its characteristic polynomial is given by
λ4 + qλ2 + 1 = 0.
(2.29)
For q > 2 the roots of (2.29) are purely imaginary. They are given by
λ = ±ia and λ = ±ib, where a = a(q) and b = b(q) are positive constants given
by
p
1
(2.30)
a2 (q) = (q + q 2 − 4)
2
and
p
1
1
b2 (q) = (q − q 2 − 4) = 2 .
(2.31)
2
a (q)
Therefore, if q > 2 then in view of the symmetry with respect to the origin,
u+ (x) can be written as
u+ (x) = 1 + A cos(ax) + B cos(bx),
(2.32)
where A and B are constants.
The function u(x), composed of u+ (x) for 0 6 x 6 ζ and u− (x) for x > ζ,
will be a solution of Problem (2.24) if we can find values for the parameters q, ζ,
A and B such that u+ and u− , as well as their first three derivatives, match at
49
x = ζ. It then follows that u ∈ C 4 ( ), and that u satisfies the Energy Identity
(2.18). This gives us the following set of conditions:
1 + A cos(aζ) + B cos(bζ)
aA sin(aζ) + bB sin(bζ)
a2 A cos(aζ) + b2 B cos(bζ)
a3 A sin(aζ) + b3 B sin(bζ)
1
,
2
1
=
µ,
2
1
= − µ2 ,
2
1
= − µ3 .
2
=
(2.33)
(2.34)
(2.35)
(2.36)
If we eliminate B from (2.34) and (2.36), we obtain
A sin(a ζ) = −
µ µ2 + b 2
,
2a a2 − b2
(2.37)
and if we eliminate A from (2.34) and (2.36), we obtain
B sin(b ζ) = +
µ µ2 + b 2
.
2b a2 − b2
(2.38)
Similarly, when we eliminate A and B from (2.33) and (2.35), we find that
1 µ2 − b 2
2 a2 − b 2
(2.39)
1 µ2 − a 2
.
2 a2 − b 2
(2.40)
A cos(a ζ) = −
and
B cos(b ζ) = +
We finally eliminate the constants A and B by dividing (2.37) by (2.39) and
(2.38) by (2.40). This yields the expressions

µ b2 + µ2 def


= −P (q),
 tan(aζ) = − 2
a b − µ2
2
2


 tan(bζ) = − µ a + µ def
= −Q(q).
b a2 − µ2
(2.41)
Lemma 2.4 The functions a(q), b(q) and µ(q) have the following properties:
50
(a) a(q) is strictly increasing, b(q) is strictly decreasing, and µ(q) is strictly
decreasing.
(b)
a(q) → 1,
b(q) → 1,
µ(q) →
q
√
2−1
as
q → 2+ .
(c)
1
√ q 1 − 2 + O(q −4 )
2q
as
q → ∞.
1
1 b(q) = √ 1 + 2 + O(q −4 )
q
2q
as
q → ∞.
1
1 µ(q) = √ 1 − 2 + O(q −4 )
q
2q
as
q → ∞.
a(q) =
(d)
(e)
Lemma 2.4 is readily proved by inspecting the expressions for µ(q), a(q) and
b(q) given by respectively (2.8), (2.30) and (2.31).
Inspection of the functions P (q) and Q(q) reveals the following:
Lemma 2.5 The functions P (q) and Q(q) have the following properties:
(a) P (q) is strictly increasing and Q(q) is strictly decreasing.
(b)
(c)
1
P (q) → p√
,
2−1
1
Q(q) → p√
2−1
o
n
3
P (q) = q 1 − 4 + O(q −8 )
2q
as
as
q → ∞.
(d)
Q(q) = 1 +
1
+ O(q −4 )
q2
51
as
q → 2+ .
q → ∞.
Lemma 2.5 is proved by an elementary explicit computation.
From (2.41) we obtain two expressions for ζ(q):
def
1 mπ − arctan P (q) ,
a(q)
def
1 nπ − arctan Q(q) ,
b(q)
ζ(q) = Xm (q) =
and
ζ(q) = Yn (q) =
(2.42)
(2.43)
where m and n are integers. Since ζ > 0, they need to be positive. Plainly, the
values of q for which a homoclinic orbit may exist must solve the equation
Xm (q) = Yn (q)
(2.44)
for some positive values of m and n.
Lemma 2.6 The functions Xm are decreasing and the functions Yn are increasing.
Proof. According to Lemma 2.4, a(q) is increasing, and according to Lemma 2.5,
P (q) is increasing, so that mπ − arctan(P (q)) is decreasing. Therefore the quotient is decreasing. Similarly, we know from Lemma 2.4 that b(q) is decreasing
and from Lemma 2.5, that Q(q) is decreasing. Therefore, nπ − arctan(Q(q)) is
increasing, and the quotient is increasing.
Note that at q = 2, we have
1
p
Xm (2) = mπ − arctan
,
√
2−1
1
,
Yn (2) = nπ − arctan p√
2−1
and for large values of q the behaviour of Xm (q) and Yn (q) is given by:
1 −1/2
Xm (q) ∼ m −
πq
as q → ∞,
2
1
Yn (q) ∼ n −
πq 1/2
as q → ∞.
4
Hence, for every pair (m, n) such that m > n, there exists precisely one point
where the graphs of Xm and Yn intersect, and therefore one root qm,n of equation
(2.44). Graphs of Xm (q) (m = 2, 3, 4, . . . ) and Y1 (q) are shown in Figure 2.4.
52
Figure 2.4: Plots of Xm (q), m = 2, 3, 4, . . . and Y1 (q)
For q to admit a homoclinic orbit to u = 0, it is necessary that it is a solution
of (2.44). However, it is not sufficient. We also require that
u(x) >
1
2
for
x ∈ (−ζ, ζ).
(2.45)
In the following lemma we show that this condition is satisfied if and only if
n = 1 and m > 2.
Lemma 2.7 For the pair (m, n), the root qm,n of equation (2.44) corresponds
to a solution of Problem (2.24) of the form (2.27) if and only if n = 1 and
m > 2.
Proof. Suppose that n > 2. Then, by (2.43),
ζ(q) >
3π
1 2π − arctan Q(q) >
.
b(q)
2b(q)
π
Thus, the point y = b(q)
lies in the interval (−ζ, ζ). We compute u at the points
π
x = 0 and y = b(q) . We have
u(0) = 1 + B + A,
u(y) = 1 − B + A cos(a2 π).
53
From (2.39), (2.40) and (2.41) we deduce that
|A|
=
|B|
=
1p
b2 (q) − µ2 (q)
1 + P 2 (q) · 2
,
2
a (q) − b2 (q)
a2 (q) − µ2 (q)
1p
.
1 + Q2 (q) · 2
2
a (q) − b2 (q)
(2.46)
(2.47)
This means in particular that
1
|B| − |A| > √
2
Thus
for every q > 2.
1
1
min{u(0), u(y)} 6 1 − |B| + |A| 6 1 − √ < .
2
2
Because we know that u(0) > 1 + √12 we conclude that u(y) < 12 , and we have
shown that for n > 2 the graph of u(x) crosses the line u = 21 at some point in
(−ζ, ζ). Therefore u(x) cannot be a solution of (2.24).
It remains to show that for n = 1 and m > 2 the point qm,n corresponds to
a solution of Problem (2.24). We shall show that u0 < 0 on the interval (0, ζ).
1
√
We observe from (2.43) that ζ < 3π
4 b q.
For convenience, we rescale the variables and define
√
y = x q,
v(y) = u(x)
and
ζ
ζ˜ = √ .
q
(2.48)
We consider the intervals I1 = (0, π4 b(q)1√q ) and I2 = ( π4 b(q)1√q , 3π
4 ) sepa00
0
0
rately: we show that v < 0 on I1 and v < 0 on I2 . Since v (0) = 0 and ζ˜ < 3π ,
this implies that v 0 < 0 on (0, ζ̃], as asserted.
For y ∈ I1 we have
√
√
v 00 (y) = −Aqa2 cos(a qy) − Bqb2 cos(b qy)
1
1
< |A|qa2 − Bqb2 √ 6 |A|qa2 −
< 0,
2
2
and for y ∈ I2 we have
v 0 (y)
√
√
√
√
= −A qa sin(a qy) − B qb sin(b qy)
1
√
√
√ 1
< 0.
< |A| qa − B qb √ 6 |A| qa(q) −
2
2
54
4
The last two inequalities follow from a simple Maple calculation.
We conclude that when q = qm,1 , where m is an arbitrary integer greater
than one, then Problem (2.24) possesses a solution u(x) of the form (2.27). In
what follows we shall put
qk = qk+1,1 ,
k > 1,
and we denote the corresponding solutions by uk (x).
2.4
Asymptotics
The asymptotic estimates obtained in Section 3 enable us to establish the asymptotic behaviour of the eigenvalues qk , the points of intersection ζk and the homoclinic orbits uk as k → ∞.
Lemma 2.8 The constants qk and ζk have the following asymptotic behaviour:
qk =
and
ζk =
=
4
2
k + + O(k −1 )
3
3
3π √
qk
4
√
π 3√
2
1−
1
2qk2
k + o(1)
as
k → ∞,
4
+ 1 + O(qk−4 )
3π
as
k → ∞.
Proof. From the two expressions (2.42) and (2.43) for ζ we obtain a relation
between qk and k. By Lemma 2.5, P (q) → ∞ and Q(q) → 1 as q → ∞. Hence,
by the asymptotic expansions of arctan(x) for x → ∞ and for x → 1, we find
that
1
a(q)
1
π
π
+ 2 + O(q −6 )
.
mπ − + + O(q −3 ) =
π−
2
q
b(q)
4
2q
Using the asymptotic expansions of a(q) and b(q) for q → ∞ from Lemma 2.4,
we obtain
4
2
−1
qm,1 = m − + O(qm,1
)
as m → ∞.
3
3
The desired estimate follows when we put m = k + 1.
55
Remark. We have
arctan(1 + x) =
π
1
1
+ x − x2 + O(x3 )
4
2
4
as x → 0.
We use equation (2.43) to obtain a relation between ζk and qk :
def
ζ(q) = Y1 (q) =
1 π − arctan Q(q) .
b(q)
Estimating b(qk ) by means of Lemma 2.4 and Q(qk ) by means of Lemma 2.5 we
obtain the estimate for ζk as k → ∞.
In order to find the asymptotic behaviour of uk (x) we first derive the behaviour of the constants Ak and Bk as k → ∞.
Lemma 2.9 The coefficients Ak and Bk have the following asymptotic behaviour:
1
as
k → ∞,
A(qk ) = (−1)k+1 3 + O(qk−5 )
qk
and
1
B(qk ) = √
2
1+
1
19
+ 4 + O(qk−6 )
2
2qk
8qk
as
k → ∞.
Proof. It follows from Lemma 2.4 and Lemma 2.8 that
b(qk ) ζk =
3π
1
− 2 + O(qk−4 )
4
2qk
as
k → ∞.
Using Lemma 2.4 to estimate the right hand side of (2.40) we find the desired
limit for B(qk ).
To obtain the limiting behaviour for A(qk ) we use the energy identity. Evaluating it at the origin we find that
Because F (1 +
1 4
a
2
√1 )
2
(
2
1 2
a A + b2 B = F (1 + A + B).
2
= 0, we can write this as
2 )2
1
b
1
=F 1+ √ +A+ B− √
B
.
A+
a
2
2
56
Expanding the left hand side and the right hand side, and using the previous
results, we obtain the desired estimate for A(qk ).
We now turn to the asymptotic behaviour of the solution uk (x). The limiting
behaviour of ζk established in Lemma 2.8 suggests to rescale the equations as
in (2.48). When we thus transform the differential equation we find that vk (y)
is a solution of the equation
1 iv
v + v 00 + f (v) = 0,
q2
and that ζ̃k →
3π
4
(2.49)
as k → ∞. We prove the following limit:
Lemma 2.10 We have
vk (y) → V (y)
as
k→∞
uniformly in C 2 ( ), where V is the unique even positive solution of the problem
V 00 (y) + f V (y) = 0,
V (±∞) = 0.
Remark. Note that

1

 1 + √ cos(y)
2
V (y) =

1
 e3π/4 e−|y|
2
for
for
3π
,
4
3π
|y| >
.
4
|y| 6
Proof. For y > ζ˜k we have
vk (y) =
1 −µk √q(y−ζ̃k )
,
e
2
so that we deduce from the asymptotic behaviour of µ(q), as given in Lemma
2.4, that
1 3π
3π
vk (y) → e 4 −y
for
y>
.
(2.50)
2
4
For y 6 ζ˜k we have
√
√
vk (y) = 1 + Ak cos a(qk ) q k y + Bk cos b(qk ) q k y .
57
(2.51)
√
√
We recall from Lemma 2.4 that a(q) q ∼ q and that b(q) q → 1 as q → ∞
1
and from Lemma 2.9 that Ak = O(k −3 ) and Bk → √2 as k → ∞, we find that
1
vk (y) → 1 + √ cos(y)
2
for
|y| 6
3π
.
4
(n)
Differentiation of (2.51) gives us that vk (y) → V (n) (y) for n = 1, 2, as asserted.
Remark. Note that the convergence is uniform in C 2 ( ), but that vk000 (y) does
(n)
not converge, even pointwise. In the Appendix we show that derivatives vk of
order n > 3, converge to V (n) in some weak sense.
2.5
Homoclinic orbits to periodic solutions
In this section we show how the solitons found in the previous sections are
embedded in families of homoclinic orbits with oscillatory tails. These tails are
associated with the imaginary eigenvalues λ = ±iω given in (2.8), and take the
form
w(x) = ε cos(ωx + φ),
(2.52)
where ε is the amplitude and φ a phase shift. Since f (s) = −s for s ∈ [− 21 , 12 ]
only, we need to restrict ε to the interval [− 12 , 12 ]. Without loss of generality we
shall always assume that ε > 0. The energy E of w defined in (2.18) is found to
be
p
1p 2
q + 4 q + q 2 + 4 ε2 .
(2.53)
E(w) = −
4
As before we look for even solutions, so that we consider the initial-boundary
value problem
 iv
00
for
x > 0,

 u + qu + f (u) = 0
0
00
u(0) = α, u (0) = 0, u (0) = β, u000 (0) = 0,
(2.54)


u(x) → w(x)
as x → ∞,
where ε has been fixed and φ will be chosen appropriately. We look for a solution
u(x) of Problem (2.54) which lies entirely above the line u = − 21 , and intersects
the line u = + 21 exactly at one point ζ ∈ + . We write
(
u+ (x)
for
0 6 x 6 ζ,
(2.55)
u(x) =
u− (x)
for
ζ 6 x < ∞.
58
Solving Problem (2.54) for u >
1
2
and for |u| <
1
2
we find that
u+ (x) = 1 + A cos(ax) + B cos(bx),
(2.56)
where A and B are constants, and
u− (x) = Ce−µx + ε cos(ωx + φ),
(2.57)
in which C is a constant. The amplitude ε will be chosen sufficiently small, so
that u(x) intersects u = 21 only once on + .
At ζ we need to match u+ and u− as well as their first three derivatives.
This yields the following conditions on the constants A, B, C, ω and φ. Because
u+ (ζ) = 12 and u− (ζ) = 12 we obtain
1
,
2
1
A cos(aζ) + B cos(bζ) = − .
2
Matching the derivatives yields:
Ce−µζ + ε cos(ωζ + φ) =
(2.58)
(2.59)
−a A sin(aζ) − b B sin(bζ) = −µ Ce−µζ − εω sin(ωζ + φ),
(2.60)
2
2
2
−µζ
2
−a A cos(aζ) − b B cos(bζ) = +µ Ce
− εω cos(ωζ + φ), (2.61)
+a3 A sin(aζ) + b3 B sin(bζ) = −µ3 Ce−µζ + εω 3 sin(ωζ + φ).
(2.62)
The system (2.58) – (2.62) can be simplified. First, the four last equations
are linear in A and B and hence these variables can be expressed in terms
of the others. In this way, we have a system with two fewer equations and
unknowns. Second, by division, it can be expressed in terms of tan(aζ) and
tan(bζ). Introducing the variable
def
θ = ωζ + φ,
(2.63)
we can write the resulting system as
Ce−µζ + ε cos(θ) =
1
,
2
(2.64)
and
a[b2 − µ2 + 2(µ2 + ω 2 )ε cos(θ)]
,
µ(b2 + µ2 )[1 − 2ε cos(θ)] + 2εω(ω 2 − b2 ) sin(θ)

b[a2 − µ2 + 2(µ2 + ω 2 )ε cos(θ)]

 cot(bζ) = −
.
µ(a2 + µ2 )[1 − 2ε cos(θ)] + 2εω(a2 − ω 2 ) sin(θ)



 cot(aζ) = −
59
(2.65)
Thus, the original system is reduced to a set of two equations with four parameters: ε, q, ζ and θ.
For ε = 0 this system is equivalent to that found in (2.41), and for ε 6= 0 we
expect to find a branch of solutions which passes through qk when ε = 0. Note
that we have written the system in a slightly different manner to (2.41). This
is due to some continuity argument that we will need.
Before exhibiting this branch we first simplify the system above still further.
This can be done thanks to the following lemma, which enables us to eliminate
the variable θ.
Lemma 2.11 Let u be a solution of equation (2.54), with the form (2.55),
where u+ and u− are given by (2.56) and (2.57) respectively, and θ is defined
by (2.63). Then we have
sin(θ) = 0.
Proof. The proof relies on following the energy E(u) along the solution. Since
it is constant, we must have
E(u+ ) = E(u− ) = E(w),
(2.66)
where E(w) is given in (2.53). An elementary computation shows that
h
i
o
p
p
−1 np 2
q − 4 q + q 2 − 4 A2 − −q + q 2 − 4 B 2 + 1
4
(2.67)
From (2.58) - (2.62) we obtain the following expressions for A and B in terms
of q, ζ, ε and θ:
b2 − µ2 + 2ε(µ2 + ω 2 ) cos θ
A=
2(a2 − b2 ) cos(aζ)
E(u+ ) =
and
B=−
a2 − µ2 + 2ε(µ2 + ω 2 ) cos θ
.
2(a2 − b2 ) cos(bζ)
When we substitute them into (2.67), we obtain the much simpler expression
p
2ε(1 − 2ε cos θ) sin θ
1p 2
p
q + 4(q + q 2 + 4)ε2 −
=
4
q2 − 4
2ε(1 − 2ε cos θ) sin θ
p
= E(u− ) −
.
q2 − 4
E(u+ ) = −
60
Thus, the equality E(u+ ) = E(u− ) implies that either ε = 0 or, since |ε| <
that sin θ = 0.
1
2,
Lemma 2.11 allows us to considerably simplify the equality (2.65). Since
the term cos θ is multiplied by ε, either we consider ε > 0 and cos θ = ±1 or
and set cos θ = 1. Thus, the
else, admit negative values of ε, so that ε ∈
bifurcation diagram is described by

a b2 − µ2 + 2(µ2 + ω 2 )ε def
1


,
= −

 cot(aζ) = −
2
2
µ(b + µ )(1 − 2ε)
P (q, ε)
(2.68)

b a2 − µ2 + 2(µ2 + ω 2 )ε def
1


.
= −
 cot(bζ) = −
µ(a2 + µ2 )(1 − 2ε)
Q(q, ε)
Note that P (q, 0) = P (q) and Q(q, 0) = Q(q). In what follows we shall drop the
bar over P and Q.
Thus, formally, in a small neighborhood of qk , the bifurcation diagram
around qk is described by the equation
Hk (q, ε) = 0,
(2.69)
where
2
Hk (q, ε) = a (q) π − arccot
1
Q(q, ε)
−(k+1)π+arccot
1
P (q, ε)
, (2.70)
for k = 1, 2 . . . .
With the assistance of Maple, it can be shown analytically that the functions
∂Hk
(q, 0)
∂q
and
q −3
∂Hk
(q, 0)
∂ε
do not depend on k, they are decreasing with respect to q for every q > 2 and
we have
3π
∂Hk
(q, 0) &
,
∂q
4
k
In particular, ∂H
∂q (qk , 0) >
the following result holds:
q −3
3π
4
∂Hk
(q, 0) & −1
∂ε
as q → ∞.
(2.71)
> 0. Thus, by the Implicit Function Theorem
Theorem 2.2
61
+
1. For every k > 1 there exists an interval [ε−
k , εk ] around ε = 0 and an
− +
analytic function qk : [εk , εk ] →
which solves (2.69) and satisfies
qk (0) = qk .
2. We have
qk−3
4
dqk
(0) %
dε
3π
as
k → ∞.
Part 2 of Theorem 2.2 follows at once from (2.71).
This is a local result. The next question is to find the maximal interval
+
[ε−
,
k εk ] along which we can extend the branches. This issue is closely related
to the behaviour of u(x) and the existence and position of its local maxima and
minima.
This discussion is made in detail in the next section. Note that we have to
−
take into acount two possibilities: at the limit values ε = ε+
k and ε = εk , either
the condition
∂Hk
qk (ε), ε 6= 0
(2.72)
∂q
fails, or either the corresponding solution of Problem (2.54) fails to verify the
condition (2.55), as it was assumed at the beginning of this section. In other
words, that the following condition fails
u(x, ε) 6=
1
2
for every
x 6= ±ζ.
(2.73)
Note that we denote the solution by u(x, ε) to emphasize its dependence with
respect to ε.
Let us first eliminate the first possibility in the following lemma
def
Lemma 2.12 Let Ω = (2, ∞) × − 12 , 21 . Then, we have
∂Hk
(q, ε) 6= 0,
∂q
for every
(q, ε) ∈ Ω.
In particular,
the functions qk (ε) defined on Theorem 2.2, can be extended to
ε ∈ − 12 , 21 .
As a consequence, we can find the solution u(·, ε) of Problem (2.54) on some
+
subinterval [ε−
k , εk ] for every k > 1. The only condition that remains to be
taken into acount is condition (2.73). Therefore, we define
def
ε+
k = sup ε > 0 : Condition (2.73) holds on (0, ε)
62
and
def
ε−
k = inf ε < 0 : Condition (2.73) holds on (ε, 0) .
In the next section we will study in more detail the dependence of u with respect
to ε for the case k = 1. We have sufficient ingredients to state the main theorem
of the section
+
Theorem 2.3 For every k > 1 and ε ∈ [ε−
k , εk ] there exists an even function
uk (·, ε) that solves the differential equation in (2.54) with q = qk (ε) and such
that
as
x → ±∞.
uk (x, ε) = ε cos ω(q) x − ζ(q, ε) + o(1)
2.6
The extended bifurcation branch
In this section we follow the Embedded Solitons along the bifurcation branches,
both for ε > 0 and for ε < 0. We first focus on the branch passing through
q1 , i.e. we set k = 1, and then we briefly describe comparable results for the
branches through the higher eigenvalues, i.e. for k > 1. We want to analyse the
dependence of the orbits on ε. For this purpose, in this section we will adopt
an approach partially analytical, partially numerical.
To emphasize the dependence of the orbit on ε, we denote it by u(x, ε). Let
us first describe what happens when ε becomes positive and when it becomes
negative. For small ε > 0 we see that the graph of u(x, ε) develops an inflection
point – and then a local maximum and a local minimum – in the region in +
where u > 12 , that is in the interval (0, ζ). Eventually, at ε = ε+ , the local
minimum touches the line u = 12 and the branch terminates. In Figure 2.5 we
show two graphs of u(x, ε), one for ε ∈ (0, ε+ ) and one for ε = ε+ .
For small ε < 0 the graph of u(x, ε) develops an inflection point – and then
a local maximum and a local minimum – in the region in + where u < 12 , that
is in the interval (ζ, ∞). As ε drops further the branch terminates at ε = ε−
when the local maximum touches the line u = 12 . In Figure 2.6 we show two
graphs of u(x, ε), one for ε ∈ (ε− , 0) and one for ε = ε− .
Summarising we find that as ε increases from zero, the graph of uk (x) first
intersects the line u = 21 at a point in the interval (0, ζ) and as ε decreases from
zero, uk (x) first intersects the line u = 21 at a point in the interval on (ζ, ∞).
Thus, as we increase ε we follow the graph of u+ (x, ε) on (0, ζ) and as we
decrease ε we follow u− (x, ε) on the interval (ζ, ∞) respectively. We begin by
63
2
2
1.5
1.5
1
1
0.5
–10
–8
–6
–4
0.5
0
–2
2
4
6
x
8
–10
10
–8
–6
–4
(a) 0 < ε < ε+
0
–2
2
4
x
6
8
10
6
8
10
(b) ε = ε+
Figure 2.5: The upper branch.
2
2
1.5
1.5
1
1
0.5
0.5
–10
–8
–6
–4
–2
0
2
4
x
6
8
–10
10
–8
–6
(a) ε− < ε < 0
–4
0
–2
2
4
x
(b) ε = ε−
Figure 2.6: The lower branch.
proving an introductory lemma which states that it suffices to consider u+ (x, ε)
and u− (x, ε) on the intervals
2π
π
I+ = ζ(q) −
, ζ(q)
and
I− = ζ(q), ζ(q) +
. (2.74)
a(q)
ω(q)
Lemma 2.13 Let ε ∈ (− 12 , 12 ). Then the following statements hold:
1. If u(x, ε) <
1
2
for every x ∈ I− , then u(x, ε) <
1
2
for |x| > ζ.
2. Suppose that q > 2.47 and − 41 6 ε 6 21 . If u(x, ε) >
then u(x, ε) > 21 for |x| < ζ.
64
1
2
for every x ∈ I+ ,
3. Let ε ∈ (0, 21 ). Then u(x, ε) <
1
2
for |x| > ζ.
Remark. Note that Part 2 of Lemma 2.13 is a statement about u(x, ε) under
the condition that q > 2.47 and − 14 6 ε 6 12 . For ε = 0, all the eigenvalues qk
verify this condition. If we follow the branches along ε ∈ (− 21 , 12 ), all of them
continue to have the property that qk (ε) > 2.47. Thus, the only restriction that
remains involves the lower bound for ε. Numerical computations show that if
we decrease ε from zero, then, before reaching the value − 41 , the bifurcation
branch is already broken. Summarising, we can state that the condition of Part
2 is always satisfied along the branches of embedded solitons.
Proof. For x > ζ, u(x, ε) is defined by (2.57). By Lemma 2.11 we can set
φ = −ωζ and (2.64) gives us the value for C. Substitution on (2.57) gives us
1
u− (x, ε) =
− ε e−µ(x−ζ) + ε cos ω(x − ζ) .
(2.75)
2
Thus u− (x, ε) is the sum of a decreasing function and a periodic one (it corresponds to the function w defined in (2.52)) that reaches its extremum at x = ζ,
which is a maximum for ε > 0 and a minimum for ε < 0. This implies that over
half a period, this is, for ζ 6 x 6 ζ + ωπ , u− reaches its global maximum in the
region [ζ, ∞). This proves Part 1 of Lemma 2.13.
If ε > 0, then ζ is indeed the global maximum. This proves Part 3 of Lemma
2.13.
In order to prove Part 2 of the lemma, we need expressions for the coefficients
A and B in u+ (x, ε). They will also be useful for explicit computations. Using
the formulae obtained in Section 5, we find
p
1 + P 2 (q, ε) b2 (q) − µ2 (q)
2
2
A(q, ε) = 2
+
ε
ω
(q)
+
µ
(q)
,
(2.76)
a (q) − b2 (q)
2
p
1 + Q2 (q, ε) a2 (q) − µ2 (q)
2
2
+
ε
ω
(q)
+
µ
(q)
B(q, ε) = 2
.
(2.77)
a (q) − b2 (q)
2
By means of an explicit computation we can check that B(q, ε) > 0 for ε ∈
[− 14 , 12 ] and q > 2.47 (in fact, it is trivial from (2.77) that B is positive for
ε > 0). As it was mentioned in the Remark above, we know a posteriori that
along the branches, ε and q lie in the assigned intervals.
We note from (2.68) that
0 < b(q(ε)ζ(q, ε)) = π − arctan(Q(q, ε)) < π,
65
so that the term cos(b(q)x) is decreasing for 0 < x < ζ. Recall that u+ (x, ε) is
2π
defined on x ∈ (0, ζ) by (2.56) and consider y = x − a(q)
. Then we have
u+ (y, ε) = 1 + A cos(ax) + B cos(by) < 1 + A cos(ax) + B cos(bx) = u + (x, ε).
Thus, the maximum of u+ is reached in the interval I+ . This completes the
proof of Lemma 2.13.
By Lemma 2.13, we are only interested in the behaviour of u(x, ε) for x ∈ I+
since for x > ζ we already know that u(x, ε) < u(ζ, ε) = 21 . We will carefully
analyse the case k = 1, the first branch, for ε > 0 by studying the behaviour of
u0 (x, ε) on I+ .
For ε = 0, the function u+ (x, 0) is decreasing. Thus, u0+ (x, 0) is negative for
x ∈ (0, ζ) and at the boundaries u0+ (0, 0) = 0 and u0+ (ζ, 0) < 0. As u00+ (0, 0) < 0,
by continuity, we can conclude that for ε > 0 sufficiently small, u0+ (x, ε) < 0 for
x ∈ (0, ζ).
Recall that u0 is given by
u0 (x, ε) = −Aa sin(ax) − Bb sin(bx).
(2.78)
With this expression we can simplify our analysis. As 0 < x < ζ < πb and
B(q, ε) > 0, the second term is always negative. By a similar argument, as
π
0 < x < ζ < 2π
a , we know that the first term is negative for 0 < x < a and
π
positive for a < x < ζ. Thus, we are only concerned about this second interval
π
a < x < ζ.
Consider now the point where the first term has an extremum, this is, the
point x = 3π
2a . Then
3πb
3π
0
, ε = Aa − Bb sin
.
(2.79)
u
2a
2a
With Maple we check that the right hand side of (2.79) is positive for ε > 0.18.
The choice of x is not optimal, but it is sufficient to show that for ε > 0 large
enough, u+ is not monotone. This motivates the following definition:
1
∗
0
ε = sup ε ∈ 0,
: u (x, ε) < 0 for 0 < x < ζ ,
(2.80)
2
x∗ = sup x ∈ (0, ζ) : u0 (x, ε∗ ) < 0 .
(2.81)
From the argument above we can conclude that 0 < ε∗ < 0.18. Indeed we can
find the values of ε∗ and x∗ as the solution of the following system in q, ε and
66
x:
H(q, ε) = 0,
u0 (x, ε) = 0,
u00 (x, ε) = 0.
(2.82)
Explicit computations show that ε∗ ≈ 0.17 and x∗ ≈ 3.33.
Thus, for 0 < ε < ε∗ , u+ is decreasing and at ε = ε∗ a point of inflection x∗
appears. We expect that for ε > ε∗ two local extrema appear in the graph of u,
namely a local minimum and a local maximum. Indeed this is the case. Let us
show that this happens.
Fix x = x∗ and consider u0+ (x∗ , ε) for ε > ε∗ , a function depending on ε
only. Computations show that this function is increasing for ε > ε∗ and hence
is positive for ε > ε∗ . As u0+ (x∗ , ε) is negative for x = πa and x = ζ, we conclude
that it has at least two zeros, one corresponding to a local minimum and one
to a maximum of u0+ (x, ε). We denote the minimum by xm (ε). We know that
xm (ε) < x∗ .
We are interested in studying the behaviour of u(xm (ε), ε) for ε > ε∗ . We
can check, again via Maple, that u(x∗ , ε) crosses the line u = 21 for ε ∈ (0, 21 ).
This implies that there exists some ε ∈ (0, 12 ) for which u(xm (ε), ε) = 21 . Thus,
we define
1
1
+
∗
: u(xm (ε), ε) >
.
(2.83)
ε = sup ε ∈ ε ,
2
2
This value of ε is the value where the bifurcation diagram breaks. It can be
found as the solution of the system
H(q, ε) = 0,
u0 (x, ε) = 0,
We find numerically that ε+ ≈ 0.19826.
67
u(x, ε) =
1
.
2
(2.84)
68
Chapter 3
Fronts on a lattice
Abstract
Motivated by a model of a system of many particles at low densities, we consider
a lattice differential equation with two uniform steady states and we investigate
the existence of travelling fronts connecting them. This leads to a two-point
boundary value problem for a nonlinear delay-differential equation. We substitute the original parabolic nonlinearity by a piece-wise linear function, where
explicit computations are possible. We find monotone and nonmonotone fronts.
Finally we also describe all the fronts such that the α-limit is the unstable
uniform state. For different values of the wave speed c of the front we find
bounded, unbounded as well as eventually periodic orbits, i.e., orbits uc (x) that
are periodic for x > xper (c) for some xper (c) ∈ .
3.1
Introduction
In this paper we study the propagation of fronts in the Cauchy problem for a sequence of functions {un (t)}∞
−∞ defined on a one-dimensional infinitely extended
lattice:
(
(3.1a)
u0n = −un + u2n−1 ,
n∈ , t>0
un (0) = an
n∈
,
(3.1b)
where an ∈ [0, 1] for every n ∈ .
Problem (3.1) arises in the context of a clock model for a dilute gas of N
particles with short range interactions, in which every particle carries a clock
69
with a discrete time k ∈ which is advanced at every collision. This happens
according to the following rule: when two particles collide, they both reset their
respective clock values, say k and `, to either k + 1 or ` + 1, whichever is the
largest. Thus, if we denote the number of particles with clock value k by Nk ,
we obtain the following dynamical equation:
k−1
∞
X
X
dNk
Rk−1,` ,
Rk,` − 2Rk,k + 2
=−
dt
`=−∞
`=−∞
`6=k
where Rk,` denotes the rate by which collisions occur between particles with
clock values k and `. We assume this rate to be proportional to Nk N` /N 2 when
k 6= ` and to Nk2 /(2N 2 ) when two particles with equal clock value k collide.
Then, writing fk = Nk /N and scaling the time appropriately, we are led to the
equation
dfk
2
2
2
= −fk + fk−1
+ 2fk−1 Ck−2 = −fk + Ck−1
− Ck−2
,
dt
where we have set
Ck =
k
X
f` .
`=−∞
Adding the equations for f` for all values of ` 6 k then yields
dCk
2
= −Ck + Ck−1
.
dt
(3.2)
For further details of this model we refer to [70]. Equations such as (3.2) also
arise in the context of cellular neural networks. For details we refer to [19] and
[38] and the references cited there.
We can place equation (3.1) in the framework of a family of classical equations by writing it as
u0n = −un + un−1 − f (un−1 ),
(3.3)
f (s) = s(1 − s).
(3.4)
wt = −wx − w(1 − w).
(3.5)
where
The first two terms on the right of (3.3) can then be viewed as a “convection”
term and we can draw an analogy between (3.3) and the first order PDE
70
It is well known that this equation has monotone travelling fronts of the form
u(x, t) = ϕ(ξ),
ξ = x − ct
c∈
,
between the two constant solutions w = 0 and w = 1 such that
ϕ(−∞) = 0
and
ϕ(+∞) = 1,
(3.6)
for wave speeds above a critical speed c∗ = 1. This warrants the conjecture
that equation (3.1) also has increasing fronts for a half-infinite interval of wave
speeds c. Such fronts would be of the form
un (t) = ϕ(ξ),
ξ = n − ct.
(3.7)
This is also observed numerically (see [26]).
A great deal of work has been done on the study of travelling waves on
discrete versions of the Fisher equation
wt = wxx − f (w),
(3.8)
in which the second derivative is replaced by the discrete Laplacian and f is given
by (3.4). It has also been studied the discretization of the Nagumo equation,
i.e., the discretized equation (3.8) with f given by the cubic function
f (s) = s(1 − s)(s − a),
for some a ∈ (0, 1).
(3.9)
We mention in particular the work by Zinner et. al. [67, 68, 69], by Hsu et.
al., [37, 38, 39, 40] and that of Mallet-Paret et. al. [18, 49], and the references
given there. However, the assumptions in there do not cover equation (3.1a).
In our first result we assume the existence of a monotone front and establish
a lower bound for the wave speed c. Specifically we prove that
c>
1
.
log 2
(3.10)
In this paper we are concerned with the existence, uniqueness and qualitative
properties of travelling fronts of equation (3.3) in which the nonlinearity f has
been replaced by a piece-wise linear function:
(
s
for s 6 21 ,
(3.11)
f (s) =
1 − s for s > 12 .
71
In this context we also mention the work of Elmer and Van Vleck, [27] and [28],
on a piecewise rendering of the Nagumo equation. For the original variable un
this yields the equation
u0n = −un + g(un−1 ),
(3.12)
where
g(s) =
(
0
for s 6 21 ,
2s − 1 for s > 12 .
(3.13)
When we look for traveling waves, and make the substitution (3.7), we obtain
u0n (t) = −c ϕ0 (ξ)
and un−1 (t) = ϕ(ξ − 1),
so that the function ϕ(ξ) needs to satisfy the delay-differential equation:
(3.14)
cϕ0 (ξ) = ϕ(ξ) − g ϕ(ξ − 1) ,
ξ∈ .
We shall seek a solution (c, ϕ) of equation (3.14) which connects the constant
states, i.e.
ϕ(−∞) = 0
and
ϕ(+∞) = 1.
(3.15)
Before stating the main existence theorem, we introduce a critical wave speed
c0 : it is the unique positive root of the equation
c e1/c = 2 e
(c0 = 4.31107 . . . ).
(3.16)
Theorem 3.1 For each c > c0 equation (3.3), with f given by (3.11) has a
unique monotone travelling wave un (t) = ϕ(n − ct), where ϕ(ξ) is a solution of
Problem (3.14), (3.15).
In Figure 3.1 we show fronts for different wave speeds c > c0 .
A natural question to ask is whether there exist fronts for speeds below the
critical speed c0 . Employing a mixture of analytical and numerical methods we
find fronts for speeds in an interval (cunb , c0 ), where cunb > 0. In this regime the
fronts are no longer monotone. In addition, we find two further critical speeds,
cunb < cper < cbif < c0 , where the shape of the front changes. In particular,
for speeds in the interval (cunb , cper ) numerical results exhibit fronts which tend
to ϕ = 0 as ξ → −∞, but which are periodic for ξ large enough. In Figure 2
we show a front for c ∈ (cbif , c0 ) (Fig 2(a)) and an eventually periodic front for
c ∈ (cunb < cper ) (Fig 2(b)).
Let us briefly describe the proof of Theorem 3.1. Since the function f given
by (3.11) is piecewise linear and the monotone travelling wave ϕ increases from
72
1
0.9
c=10
c=30
c=50
c=4.4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-60
-40
-20
0
20
40
60
Figure 3.1: Monotone fronts for some values of c > c0
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-20
-15
-10
-5
0
5
10
15
20
(a) Damped oscillation for c = 2
0
-20
-15
-10
-5
0
5
10
15
20
(b) Eventually periodic orbit for c = 1.5
Figure 3.2: Graphs of orbits for different values of c
0 to 1, there is no loss of generality when we fix ϕ(0) = 21 . Then, for ξ < 1,
the function ϕ(ξ) satisfies a linear ODE, which we can solve explicitly, and for
ξ > 1 it satisfies a linear DDE which we will analyse in detail using the Laplace
Transform. Theorem 3.1 is then proved by means of an analysis of ϕ0 (ξ).
The plan of the paper is the following: in Section 2 we first prove the lower
73
bound (3.10). In Section 3 we give a detailed outline of the proof Theorem 3.1.
The roots of the characteristic equation of the linearised equation at u = 1 will
play a central role in the analysis; this equation will be discussed in Section 4.
In Section 5 we discuss the corresponding fundamental solution. In Section 6
we prove a monotonicity property when c > c0 , and in Section 7 we wrap up
the proof of Theorem 3.1 and we present an explicit expression for the front for
c > c0 .
The existence of waves established by Theorem 3.1 is the starting point
of a study of fronts for c < c0 . This is done in Section 8. Using analytical
arguments and numerical computations, we discuss the behaviour of all the
fronts for 0 < c < c0 which satisfy equation (3.3), such that ϕ(−∞) = 0. There
we show the existence of different types of orbits: nonmonotone fronts such
that ϕ(∞) = 1, as in figure 3.2(a), bounded orbits where ϕ(∞) is not defined,
eventually periodic orbits, as in figure 3.2(b) and unbounded orbits.
3.2
A lower bound for the wave speed c
In this section we consider increasing travelling wave solutions of the equation
u0n = −un + u2n−1 ,
and establish a lower bound for the wave speed. As explained in the Introduction
we therefore need to study the problem
 0
for x ∈ ,
 cu (x) = u(x) − u2 (x − 1)
(3.17a)
1
 u(−∞) = 0, u(0) = , u(+∞) = 1,
(3.17b)
2
where we have eliminated the translation invariance by pinning the solution at
the origin.
We first prove a few bounds and estimates for increasing solutions of Problem
(3.17).
Theorem 3.2 Let u be an increasing solution of Problem (3.17). Then
1.
u(x) 6
ex/c
1 + ex/c
and hence u(x) = O(ex/c ) as x → −∞.
74
for
x < 0,
2.
1 − u(x) 6
1
1 + ex/c
for
x > 0,
and hence 1 − u(x) = O(e−x/c ) as x → +∞.
3.
def
lim u(x) e−x/c = ` ∈ ( 21 , 1).
x→−∞
Proof. The main ingredients of the proof are the monotonicity and the bounds
0 < u < 1. From the differential equation we obtain
cu0 (x) = u(x) − u2 (x − 1) > u(x) − u2 (x).
Let y > x. Then, integrating this inequality over the interval (x, y) yields
u(x)
u(y)
e−y/c >
e−x/c .
1 − u(y)
1 − u(x)
(3.18)
For x < 0 and y = 0 we obtain Part 1 and for x = 0 and y > 0 we obtain Part 2.
To prove Part 3, we transform to the variable v(x) = u(x)e−x/c . Plainly,
1
1
v 0 (x) = {u0 (x) − u(x)} e−x/c = − u2 (x − 1) e−x/c < 0.
c
c
Hence, v is a decreasing function. By the first inequality v is bounded above by
1. Therefore
1
lim v(x) = `1 > v(0) = .
x→−∞
2
Next, we define the function
w(x) =
u(x)
e−x/c .
1 − u(x)
We have shown in (3.18) that w is a strictly increasing function. Therefore
lim w(x) = `2 < w(0) = 1,
x→−∞
and
lim v(x) = lim w(x) < 1,
x→−∞
x→−∞
as asserted in Part 3.
We are now ready to prove the lower bound for the wave speed.
75
Theorem 3.3 Let u be an increasing solution of Problem (3.17). Then
c>
1
.
log 2
Proof. We return to the function v(x) = u(x)e−x/c . Because u < 1, we have
the upper bound
(3.19)
v(x) < e−x/c
for
x∈ .
An elementary computation shows that v is a solution of the integral equation
Z x
1 (t−2)/c 2
1
v(x) = −
v (t − 1)dt,
(3.20)
e
2
0 c
and that it is bounded above by


1




 1 e−x/c
v+ (x) = 2
1


2



e−x/c
x 6 −c log 2
−c log 2 < x < 0
0 6 x 6 c log 2
x > c log 2.
When we substitute this upper bound for v into the integral equation (3.20),
we obtain the following lower bound on the half line x > 1 + c log 2:
v(x) > e−x/c +
1 3 −1/c
− e
+ e−2/c .
2 2
(3.21)
From (3.19) and (3.21) we conclude that
1 3 −1/c
− e
+ e−2/c < 0,
2 2
which implies that c >
3.3
1
log 2 .
This completes the proof of the theorem.
Outline
In order to prove Theorem 3.1 we need to find a strictly increasing solution of
the two-point boundary value problem
 0
1
(3.22a)
x∈ ,

 u (x) = c u(x) − g u(x − 1) ,
u(−∞) = 0,
(3.22b)


(3.22c)
u(+∞) = 1.
76
Since u is strictly increasing from 0 to 1, there exists a unique value of x, where
its graph crosses the line u = 12 . Without loss of generality we may choose this
value to be the origin. Then u(x − 1) < 12 for x < 1, and we find that u(x) must
satisfy
 0
for − ∞ < x < 1,
 u (x) = 1c u(x)
(3.23a)
 u(−∞) = 0 and u(0) = 1 .
(3.23b)
2
Therefore
u(x) =
1 x/c
e
2
for
− ∞ < x < 1.
(3.24)
Since it is required that u(x) → 0 as x → −∞, it is evident that c needs to be
positive.
For x > 1 we have u(x − 1) > 21 and we must solve the Problem
 0
1

 u (x) = c u(x) + 1 − 2u(x − 1)
u(x) = 21 ex/c ,


u(∞) = 1.
for 1 < x < ∞,
for 0 6 x 6 1,
(3.25a)
(3.25b)
(3.25c)
It will be convenient to transform this problem, and introduce the variables
t=x−1
and
y(t) = 1 − u(x).
Problem (3.25) then becomes
 0
1

 y (t) = c y(t) − 2y(t − 1)
y(t) = η(t)


y(∞) = 0,
for 0 < t < ∞,
for − 1 6 t 6 0,
where
def
η(t) = 1 − u(1 + t) = 1 −
1 (1+t)/c
e
.
2
(3.26)
(3.27a)
(3.27b)
(3.27c)
(3.28)
We now need to identify values of c > 0 for which Problem (3.27) has a
solution with the property
y(t) <
Then u(x − 1) >
1
2
1
2
for 0 < t < ∞.
for x > 1, as assumed at the outset.
77
(3.29)
Problem (3.27) is a linear problem, which can be solved by means of the
Laplace Transform. Thus, we write formally
Z ∞
def
ŷ(s) = L(y)(s) =
y(t) e−st dt.
0
Transformation of equation (3.27a) then yields
Z
1
2 ∞
{sŷ(s) − y(0)} = ŷ(s) −
y(t − 1) e−st dt.
c
c 0
Z
Z
2 1
1
2 ∞
= ŷ(s) −
y(t − 1) e−st dt −
y(t − 1) e−st dt
c
c 0
c 1
Z
2 ∞
2
1
ψ(t) e−st dt − e−s ŷ(s),
= ŷ(s) −
c
c 0
c
where
ψ(t) =
Thus,
2
c
Since
(
η(t − 1) = 1 − et/c ,
0<t<1
0,
otherwise.
e−s + s −
1
2
ŷ(s) = y(0) − ψ̂(s).
c
c
(3.30)
(3.31)
1
y(0) = η(0) = 1 − e1/c ,
2
(3.32)
1
2
ŷ(s) = 1 − e1/c Γ̂(s) − ψ̂(s)Γ̂(s),
2
c
(3.33)
this yields the expression
where
Γ̂(s) =
2 −s
ce
1
.
+ s − 1c
(3.34)
Assuming for the moment that we can carry out the inverse Laplace Transform,
we formally obtain for y(t):
1
2
y(t) = 1 − e1/c Γ(t) − (Γ ∗ ψ)(t),
(3.35)
2
c
where ∗ denotes the convolution product
Z t
(Γ ∗ ψ)(t) =
ψ(s) Γ(t − s) ds.
0
78
If it can be shown that y(t) is a decreasing function, then substitution into
(3.26) yields an increasing solution u(x) of Problem (3.22).
3.4
The characteristic equation
The Laplace Transform Γ̂(s) of the fundamental solution Γ(t) has poles at the
zeros of the characteristic equation:
1
2 −λ
e + λ − = 0.
c
c
(3.36)
In this section we investigate these zeros. Note that if λ is a root of equation
(3.36) then so is its complex conjugate λ. Hence, we need only consider zeros
with nonnegative imaginary part.
The complex equation (3.36) for λ = a + ib is equivalent to the following
system
(
2 −a
cos b = 1c − a,
(3.37a)
c e
2
c
e−a sin b = b.
(3.37b)
We note that (3.37b) is trivially satisfied for b = 0. For a we then obtain from
(3.37a) the equation
1
2 −a
e + a − = 0.
(3.38)
c
c
An elementary analysis leads to the following theorem about the real roots of
equation (3.36):
Theorem 3.4 There exist two constant c− ∈ (0, 2) and c0 > 2 satisfying
c e1/c = 2e
(c0 = 4.31107...,
c− = 0.37336...)
such that
(a) For c < 0, equation (3.38) has exactly one positive solution of multiplicity
one.
(b) For c = 0, equation (3.38) has exactly one positive solution (λ = log 2) of
multiplicity two.
(c) For 0 < c < c− , equation (3.38) has exactly two positive solutions.
79
(d) For c = c− , equation (3.38) has exactly one positive solution of multiplicity
two.
(e) For c− < c < c0 , equation (3.38) has no real solution.
(f ) For c = c0 , equation (3.38) has exactly one negative solution of multiplicity
two:
c 1
0
a0 = −1 +
(a0 = −0.768....).
(3.39)
= − log
c0
2
(g) For c > c0 , equation (3.38) has exactly two negative solutions a1 (c) and
a2 (c) which verify a2 (c) < a0 < a1 (c) < 0, and
1
1
a1 (c) = −
as c → ∞.
+O 3
c−2
c
In the next theorem we discuss the location of the complex roots of the
characteristic equation. We make two preliminary observations:
(a) From (3.37b) we deduce the inequality
ea =
2
2 sin b
< .
c b
c
Hence, for c > 2 all the complex solutions have negative real part. Actually,
this lower bound can be improved.
Definition. The critical wave speed cbif will be defined as the biggest value for
which the characteristic equation has purely imaginary roots.
We see from (3.37) that
√
3 3
cbif =
≈ 1.654.
(3.40)
π
In this section, as well as in Sections 5, 6 and 7, we will always consider c > cbif .
In Section 8 we shall explore the range c ∈ (0, cbif ).
(b) Since by assumption, c > 0 and a < 0, it follows from (3.37a) that cos(b) > 0
and from (3.37b) that sin(b) > 0. Hence
π
b ∈ Ik = 2(k − 2)π, 2(k − 2)π +
for some k ∈ {2, 3, 4...}.
2
Elementary analysis shows that for k = 2, there exists a nonreal root iff c < c 0 .
80
Theorem 3.5 For each k > 3 there exists a unique nonreal complex root λ k =
(ak , bk ) of equation (3.36) with positive imaginary part such that if c > c0 , then
ak (c) < a2 (c) < 0, where a2 (c) is the smallest real solution of (3.36) found in
Theorem 3.4, and
π
− log 2(k − 2)π +
− log (sin bk ) < ak + log
2
2(k − 2)π <
bk
c
2
< − log 2(k − 2)π ,
< 2(k − 2)π +
π
2
for k = 3, 4, 5, . . . In particular ak = O(− log k) as k → ∞.
50
50
0
0
0
−50
y
50
y
y
In Figure 3.3 we show the diagram of the eigenvalues of equation (3.36) for
different values of wave speed c.
−50
−4
−3
−2
−1
0
−50
−4
−3
−2
x
(a) c = 5
0
−4
(b) c = c0
20
10
10
0
0
−10
y
20
10
−10
−20
−3
−1
x
(d) c = cbif
0
1
−3
−2
x
−1
0
0
−10
−20
−2
−3
(c) c = 2
20
y
y
−1
x
−20
−2
−1
x
(e) c = 1.5
0
1
−3
−2
−1
x
(f) c = 1
Figure 3.3: Location of the eigenvalues for different values of c.
81
0
1
3.5
The expansion of the fundamental solution
In the last section we have investigated the poles of Γ̂(s) that we will need, in
order to invert the Laplace operator. The inversion formula is given by
1
Γ(t) =
2πi
lim
R→∞
Z
iR
est Γ̂(s)ds
(3.41)
−iR
if est Γ̂(s) is integrable. In Lemma 3.1 it is shown that we may apply the residues
formula, and express the integral in (3.41) as a series of residues computed at
the poles of Γ̂(s), i.e. at the zeros (ak , ±bk ) of the characteristic function
h(s) =
1
2 −s
e +s− .
c
c
For convenience, let us introduce the following notation: for each c > 0,
we will denote the set of all the roots of the characteristic function h(s) by
{zk = zk (c) : k = 1, 2, . . .}. In last section we have found that they correspond
to
{a1 , a2 , ak ± ibk : k = 3, 4, . . .}
{ak ± ibk : k = 2, 3, . . .}
for c > c0 ,
for c < c0 .
Lemma 3.1 Let {zk } be the sequence of all the roots of the function h. We
have
∞
X
Γ(t) =
Res(h−1 (s), s = zk ) ezk t .
(3.42)
k=1
Proof. We consider the sequence of semicircles Sn of radius Rn :
n
π
3π o
Sn = Rn eiθ : 6 θ 6
.
2
2
Here Rn → ∞ as n → ∞. In order to prove (3.42) we need to check that the
integral taken along Sn tends to zero when n → ∞.
From Theorem 3.5 we know that bk ∈ 2(k − 3)π, 2(k − 3)π + π2 , k =
3, 4, 5 . . . . By choosing the radii Rk appropriately, we can ensure that on the
82
semicircles Sk the denominator of the integrand h(s) is bounded away from zero.
Specifically, we choose Rk = (2k + 1)π. Then
p
|Rk − λk | > Rk − |λk | = (2k + 1)π − |bk | 1 + θk ,
where
θk =
Therefore
log k 2
a2k
∼
2
bk
2πk
as k → ∞.
lim inf |Rk − λk | >
k→∞
π
.
2
With this choice of radii Rn , and with n large enough, each successive
semidisk contains one more zero and on Sn the denominator is bounded away
from zero as n → ∞.
The next step is to estimate the integrand on Sn and to show that it tends
to zero as n → ∞ sufficiently fast. To that end we divide Sn into two parts:
Sn1 = {s ∈ Sn : Re(s) 6 −µ log |s|},
Sn2 = {s ∈ Sn : Re(s) > −µ log |s|},
where µ is a positive constant.
We take µ > 1. Then on Sn1 we have
|ses | 6 e(1−µ) log |s| → 0 as |s| → ∞,
so that
|e−s h−1 (s)| =
c
c
→
|2 + cses − es |
2
as |s| → ∞.
Therefore,
sup |h−1 (s)| ∼
1
s∈Sn
c
c
sup |es | = e−µ log Rn
2 s∈Sn1
2
as Rn → ∞.
Since the length of Sn1 is less than πRn , and |est | 6 1 on Sn1 , it follows that
Z
as Rn → ∞.
h−1 (s)est ds 6 O e(1−µ) log Rn
1
Sn
Because µ > 1 the integral over Sn1 converges to zero uniformly for t > 0.
83
Next, we show that the integral over Sn2 converges to zero as well. We write
Sn2 in complex notation:
n
π
log(Rn ) o
.
Sn2 = Rn e±iθ : < θ < arccos −µ
2
Rn
The Taylor expansion
arccos(x) =
π
− x + O(x3 ) as x → 0
2
implies that, as Rn → ∞, the length of Sn2 is given asymptotically by
|Sn2 | = 2µ log(Rn ) + O
log3 (R ) n
.
2
Rn
Since the exponential factor is bounded in modulus by one, and h−1 (s) =
O(Rn−1 ) on Sn2 , we conclude that
Z
log R n
h−1 (s)est ds 6 O
as Rn → ∞.
R
2
n
Sn
Thus, we have shown that
lim
Rn →∞
Z
est Γ̂(s)ds = 0
Sn
and we can conclude by the residues formula that
1
Γ(t) =
2πi
lim
R→∞
Z
iR
est Γ̂(s)ds =
−iR
∞
X
Res(h−1 (s), s = zk )ezk t .
k=1
Since we have shown the validity of the residues formula, next we will compute the residues. We note that if c > c0 , then all the roots of h(s) are simple
and with negative real part and if c = c0 , then h(s) has one double root. We
will treat both cases separately.
Lemma 3.2 Suppose that c > c0 . Then
Γ(t) =
2
X
k=1
∞
X 1
1
ak t
λk t
e
+
e
+
cc
.
h0 (ak )
h0 (λk )
k=3
84
(3.43)
Since
1
h0 (λk ) = λk + 1 − ,
c
the expansion (3.43) can be written as
Γ(t) =
2
X
e ak t
a +1−
k=1 k
1
c
+2
∞
X
(ak + 1 − 1c ) cos(bk t) + bk sin(bk t) ak t
e . (3.44)
(ak + 1 − 1c )2 + b2k
k=3
Proof. The lemma is an immediate consequence of the residues formula (3.42).
1
1
are given by h0 (λ
.
When c > c0 the residues of Γ̂(s) = h(s)
k)
Lemma 3.3 Suppose that c = c0 . Then
∞
1 a0 t X 1
e +
eλk t + cc ,
Γ(t) = 2 t +
0
3
h (λk )
(3.45)
k=3
and the expansion (3.45) can be written as
∞
X
(ak + 1 − c10 ) cos(bk t) + bk sin(bk t) a t
1 a0 t
Γ(t) = 2 t +
e +2
e k .
3
(ak + 1 − c10 )2 + b2k
(3.46)
k=3
Proof. When c = c0 , the computation made for the zeros λk which are not real
valued is still valid since these zeros are all simple. However, in this case the
zero a0 (see Theorem 3.4) has multiplicity two and the computation is different.
Since h is analytic we have the Taylor expansion
h(s) = α0 (s − a0 )2 + α1 (s − a0 )3 + · · ·
(3.47)
which is valid in the disk with circumference Sr = {|s − a0 | = r} for some
sufficiently small r, and α0 6= 0. Therefore
1
1
α1
1
1
=
− 2
+ analytic function
2
h(s)
α0 (s − a0 )
α0 s − a 0
and, writing w = s − a0 ,
Z
Z
Z
1
est ds
ewt ds
e a0 t
e a0 t 1
α1 1
1
ewt 2 −
=
=
+ · · · ds
2πi Sr h(s)
2πi Sr h(s)
2πi α0 Sr
w
α0 w
Z a0 t
e
1
1
t
α1 1
=
+ −
+ · · · ds
2πi α0 Sr w2
w α0 w
1 α1 a0 t
=
t−
e .
(3.48)
α0
α0
85
It remains to compute α0 and α1 . We find:
α0 = lim
s→a0
and
α1 = lim
s→a0
h(s)
1
1
= h00 (a0 ) = ,
(s − a0 )2
2
2
1
1
h(s) − a0 (s − a0 )2
= h000 (a0 ) = − .
(s − a0 )3
6
6
If we use these expressions in (3.48) we obtain the desired expansions (3.45) and
(3.46).
3.6
Monotonicity
We recall that we are interested in establishing monotonicity for two reasons.
First, from the original application, un denotes the sum of positive quantities,
so it must be nondecreasing. On the other hand, it remains is to check whether
the solution u(x) given in (3.35), with Γ(t) given by the expansion formula in
(3.43) or (3.45), satisfies the assumption thatu(x) > 21 for x > 0, which we made
at the outset. Monotonicity of u(x) automatically ensures that this is indeed
the case.
As in Section 3, we transform to the variables t and y introduced in (3.26):
t=x−1
and
y(t) = 1 − u(x).
We shall prove the following monotonicity result.
Theorem 3.6 For every c > c0 the solution y(t) of the Problem
( 0
for 0 < t < ∞,
y (t) = 1c {y(t) − 2y(t − 1)}
y(t) = η(t) = 1 −
1 (1+t)/c
2e
for
t 6 0.
(3.49a)
(3.49b)
is strictly decreasing.
Proof. The proof is carried out again by means of the Laplace Transform, as in
Section 3. We begin with a preliminary result.
86
Lemma 3.4 Let y(t) be the solution of Problem (3.49). Then
y 0 (t) = −
1
Γ(t + 1)
2c
for
t > 0.
(3.50)
−st
Proof. We multiply equation (3.49a)
R ∞by e , integrate over t ∈ (0, ∞), and
express all the integrals in terms of −1 e−st y 0 (t)dt. After an elementary computation this yields
Z ∞
1 s
h(s)
e .
e−st y 0 (t)dt = −
2c
−1
Since
Z
∞
e
−st 0
y (t)dt =
−1
Z
0
−1
e−st η 0 (t)dt + L(y 0 )(s),
we obtain for the Laplace Transform of y 0
Z 0
1
1 es
L(y 0 )(s) = −
−
e−st η 0 (t)dt = − I1 (s) − I2 (s).
2c h(s)
2c
−1
An elementary computation yields for I2 (s):
Z 0
1 es − e(1/c)
I2 (s) =
e−st η 0 (t)dt = −
.
2c s − (1/c)
−1
(3.51)
(3.52)
To find an expression for I1 (s), we compute the Laplace Transform of Γ(t + 1)
Z ∞
Z ∞ Z 1 Γ(τ )e−sτ dτ.
Γ(t + 1)e−st dt = es
L(Γ(t + 1))(s) =
−
0
0
0
From [35], pages 19 and 20, we know that Γ(t) is the solution of equation (3.49a)
with initial data on [−1, 0] given by
y(t) = 0 for
− 1 6 t < 0,
and y(0) = 1.
Therefore, Γ(t) = et/c for 0 < t < 1, and hence
L(Γ(1 + t)(s) =
es
es − e1/c
es
−
=
+ 2cI2 (s).
h(s) s − (1/c)
h(s)
Thus, putting (3.52) and (3.53) into (3.51) we can conclude
L y 0 (·) (s) = L Γ(1 + ·) (s)
87
(3.53)
(3.54)
and (3.50) follows.
Thus, it suffices to show that Γ(t) is positive for t > 1. We will again treat
the cases c > c0 and c = c0 separately.
(a) Positivity of Γ(t) for the case c > c0
We recall from (3.43) that Γ(t) is given by
Γ(t) =
∞
X
1
e a2 t
e a1 t
Ak (t) 0
+
+2
e ak t ,
a1 + 1 − 1/c a2 + 1 − 1/c
|h (λk )|
k=3
where a1 and a2 are such that a1 + 1 − 1/c > 0 and a2 + 1 − 1/c < 0, and
|Ak (t)| 6 1 for every k > 3. To show that Γ(t) is positive we need a bound on
the third term R(t). By Theorem 3.5, the exponent ak t can be estimated by
Hence
c
c
sin bk
ak t = −t log
+ t log
− t log 2(k − 2)π .
6 −t log
2
bk
2
|R(t)| 6 2
=
∞
X
k=3
∞
X
1
1
e ak t 6
e−t log(c/2)−t log(2(k−2)π)
2(k − 2)π
(k − 2)π
k=3
1 −t log(πc)
e
π
∞ X
k=3
1 t+1
.
k−2
We are interested in the range t > 1. Then
|R(t)| 6
∞
π
1 −t log(πc) X 1 2
e
= e−t log(πc) ,
π
k−2
6
k=3
so that
Γ(t) >
e a1 t
e a2 t
π
def
+
− e−t log(πc) = Q(t, c)
a1 + 1 − 1/c a2 + 1 − 1/c
6
Proposition 3.1 If c > c0 , then Q(t, c) > 0 for t > 1.
88
for t > 1.
Proof. We consider the equivalent problem of showing the positivity of the
function Q(t, c)e−a1 t , i.e. of
Q(t, c)e−a1 t =
1
e(a2 −a1 )t
π
+
− e−t(a1 +log(πc))
a1 + 1 − 1/c a2 + 1 − 1/c
6
for c > c0 and t > 1. This function is seen to be increasing with respect to t.
Therefore, its minimum is reached at t = 1. Hence, it remains to show that the
function
Q(1, c) =
1
ea2 −a1
π
+
− e−(a1 +log(πc))
a1 + 1 − 1/c a2 + 1 − 1/c
6
is positive for every c > c0 . This can easily be checked by the use of a computer
program.
(b) Positivity of Γ(t) for the case c = c0
We first consider the range t > 1. There
1 a0 t π −t log(πc0 ) def
e − e
Γ(t) > 2 t +
= Q0 (t).
3
6
As in the previous case, the function e−a0 t Q0 (t) is seen to be increasing for
1 6 t < ∞, and hence e−a0 t Q0 (t) > e−a0 Q0 (1) > 0. Thus Γ(t) > 0, and we
conclude that y(t) is a decreasing function for t > 0.
3.7
The final solution
The solution of Problem (3.22) can be expressed in the form of an expansion.
Theorem 3.7 If c > c0 , the solution u(x) of Problem (3.22) can be written as

1 x/c


for x 6 1,
2 e
∞
(3.55)
u(x) =
1X
1

ezk x for x > 1.

1 − 2
(−z )(c + cz − 1)
k=1
k
k
where {zk : k = 1, 2, 3, . . . } is the set of roots of the characteristic equation
(3.36).
89
If c = c0 we have

1 x/c0



2e
∞
u(x) =
a0 x 1 X
1
1


e zk x
x
−
b
e
−
1
−


2
2
(−zk )(c0 + c0 zk − 1)
for
x 6 1,
for
x > 1,
k=2
(3.56)
where {zk } denotes all the nonreal complex roots of the characteristic function
h(s) and
3c2 + c0 − 1
b= 0
≈ 1.796
3(c0 − 1)2
Proof. Recall the change of variables used in (3.26). One way is to express the
solution is via the equality (3.35). Another possibility is to use the equality
(3.50) obtained in Lemma 3.4. We will use the latter.
Thus, in (3.50) we have transformed the original delay-differential equation
(3.27) into an ordinary differential equation


y 0 (t) = − 1 Γ(t + 1),
for 0 < t < ∞,
2c
(3.57)
1

y(0) = η(0) = 1 − e1/c .
2
The solution of (3.57) is given formally by
Z t
1
Γ(s + 1) ds
y(t) = η(0) −
2c 0
for t > 0.
(3.58)
We make use of the expansion for Γ obtained in (3.43) and (3.45) for the
cases c > c0 and c = c0 respectively.
The expansion for the case c > c0 .
We recall from (3.43) that Γ(t) was given by
Γ(t) =
∞
X
k=1
∞
X
1
e zk t =
0
h (zk )
1
k=1 c
1
e zk t .
− 1 − zk
We substitute this expression into (3.58). An elementary computation shows
that
∞
X
y(t) = η(0) +
αk (ezk t − 1),
(3.59)
k=1
90
where the constants αk are given by
αk =
1
.
zk (1 − czk )(1 − c − czk )
(3.60)
We can simplify expression (3.59). We know that there exist a limit of (3.59)
as t tends to infinity, which can only be a steady state solution of (3.27a), this
is, zero. This means
∞
X
1
αk = η(0) = 1 − e1/c
(3.61)
2
k=1
so that (3.59) becomes
y(t) =
∞
X
αk e zk t .
(3.62)
k=1
Returning to the original variables x and u(x) we obtain (3.55) as required.
The expansion for the case c = c0 .
We recall from (3.45) that Γ(t) is given by
∞
1 a0 t X 1
e +
e zk t .
Γ(t) = 2 t +
0
3
h (zk )
k=2
When we put this expression into (3.59), with c = c0 , we obtain the solution y
of Problem (3.57):
y(t) = η(0)+
∞
X
7c0 − 4 a0 t
1
7c0 − 4
t−
e +
+
αk (ezk t −1) . (3.63)
2
2
c0
3(c0 − 1)
3c0 (c0 − 1)
k=2
We can simplify this expression (3.63) for y(t). We take the limit as t tends to
∞, which will be zero. This yields the relation
∞
X
αk = η(0) +
k=2
e
7c0 − 4
7c0 − 4
=1−
+
,
2
3c0 (c0 − 1)
c0
3c0 (c0 − 1)2
(3.64)
and (3.63) becomes
y(t) =
∞
1
7c0 − 4 a0 t X
αk e zk t .
e +
t−
c0
3(c0 − 1)2
k=2
Return to the original variables x and u(x) yields (3.7).
91
(3.65)
3.8
The range c < c0
Having established the existence of monotone fronts for c > c0 , we now turn
to the regime c < c0 . In describing the different types of fronts we observe, we
distinguish a series of critical wave speeds:
c0 : The smallest value of c for which the characteristic equation (3.36) has real
roots: two for c > c0 and a double root for c = c0 (cf. Theorem 3.4).
cbif : The value of c for which the set of roots of the characteristic equation
reaches the imaginary axis (cf. (3.40)).
For cbif < c < c0 the roots still have negative real part and the uniform state
u = 1 is asymptotically stable. Thus, for cbif < c < c0 we may still expect fronts
to exist, but with oscillatory tails as x → ∞.
In the range 0 < c < cbif , we distinguish two further critical wave speeds:
cunb , below which fronts are unbounded, and cper : for c ∈ (cunb , cper ) solutions
are eventually periodic. These values are ordered according to:
0 < cunb < cper < cbif < c0 .
The numerical values are found to be
cunb = 1.455 . . . ,
cper = 1.618 . . . ,
cbif = 1.654 . . . ,
c0 = 4.311 . . . .
Our approach will be partly numerical and partly analytical. Results based in
part or wholly on numerical simulations will be formulated in Propositions.
Let us give an overview of the results that we will describe in this section.
The goal is to obtain a qualitative description of the solution u(x, c) of the
problem
 0
1
if x > 1 and u(x − 1) 6 21 , (3.66a)

 u (x) = c u(x),
if x > 1 and u(x − 1) > 12 , (3.66b)
u0 (x) = 1c {u(x) + 1 − 2u(x − 1)}

x

(3.66c)
u(x) = 21 e c
for x 6 1,
when c takes on different positive values, and we find the following:
1. If c > c0 , then u(x, c) is strictly increasing and u(∞, c) = 1.
2. If cbif < c < c0 , then u(x, c) is nonmonotone and u(∞, c) = 1.
92
3. If cunb < c 6 cbif , then u(·, c) > 0 on and bounded above. Moreover, if
cunb < c 6 cper , then there exists xper (c) 6 0 and T (c) > 0 such that
u(x, c) = u(x + T (c), c)
for every
x > xper (c).
4. If c = cunb , then there exists a constant x̃ > −1 such that
u(·, cunb ) > 0 on (−∞, x̃)
and u(·, cunb ) = 0 on [x̃, ∞).
5. If 0 < c < cunb , then
lim u(x, c) = −∞.
x→∞
Part 1 has been proved in the previous sections.
The range cbif < c < c0 .
For c < c0 , but close to c0 , small damped oscillations around the uniform state
u = 1 appear. However the solution remains above the value u = 21 for all x > 1.
As we reduce c further, we find that the amplitude of the oscillations increases
and there exists a value c1 ∈ (cbif , c0 ) for which the graph of u touches the line
u = 21 (c1 ≈ 1.664). Specifically,
1. For c1 < c < c0 we find a nonmonotone solution of the delay-differential
equation (3.66) which tends to 1 as x → ∞, such that u(x, c) > 12 for
every x > 1.
2. For c = c1 we find a nonmonotone solution of the delay-differential equation (3.66) converging to 1 and a point x1 > 1 (x1 ≈ 5.667) such that
u(x1 , c1 ) = 12 and u(x, c1 ) > 21 for every x > 1, x 6= x1 .
3. For c < c1 the condition u(x, c) >
1
2
does not hold for some x ∈ [1, ∞).
A natural question is: how much smaller can we choose c for there to exist
a front type solution, i.e. a heteroclinic orbit which connects u = 0 to u = 1.
Notice that the difference between both c1 and cbif is small (c1 − cbif ≈ 0.01);
thus, the regime where the question remains open is rather narrow. For values of
c close to cbif , the amplitude of the oscillations around u = 1 is seen to increase
and the number of times that u(x, c) crosses the line u = 21 also increases.
Motivated by numerical experiments, we formulate the following conjecture:
Conjecture 3.1
93
2
c=1.4
c=c_unb
c=1.5
c=c_per
c=c_bif
c=2
c=5
1.5
1
0.5
0
-0.5
-4
-2
0
2
4
6
8
10
12
14
Figure 3.4: Fronts for different values of c
1. There exists a sequence of values c1 > c2 > · · · > cn > · · · > cbif such
that for cn > c > cn+1 , the graph of the solution u crosses the line u = 21
transversally, exactly 2n times. For c = cn , u crosses the line u = 21
transversally, exactly 2(n − 1) times and once it is tangent to the line.
2. The sequence of values cn verifies that
def
c∞ = lim cn = cbif .
n→∞
3. For cunb < c 6 c∞ , the number of points where the orbit u(x, c) crosses
the line u = 21 is infinity. Each time the crossing is transversal.
Remark. Conjecture 3.1 implies that for every c > cbif , Problem (3.66) has a
front type solution, because there exists a point xc > 1 such that u(·, c) solves
the linear DDE (3.66b) on (xc , ∞), and hence
lim u(x, c) = 1.
x→∞
(3.67)
Part 3 of the Conjecture means that the set of values of c for which the boundary
condition (3.67) holds is precisely c > c∞ and by Part 2, c∞ = cbif .
The range 0 < c 6 cbif . Numerical experiments suggest that the solution is
bounded for cunb < c < cbif and unbounded for 0 < c < cunb . For speeds in
94
the range of bounded solutions we find another critical speed cper such that for
cunb < c 6 cper , bounded orbits are eventually periodic, i.e., there exists a point
xper (c) such that u(·, c) is periodic for x > xper (c).
In the following lemma we show how positivity and boundedness are connected. Let u(x, c) be the solution of Problem (8.1), and let
def
ξ(c) = sup{x > 1 : u(·, c) > 0 on (−∞, x)}.
(3.68)
Plainly, since u > 0 on − , ξ is well defined. If u(x, c) > 0 for all x ∈ , then
ξ(c) = ∞. We shall show that if ξ(c) < ∞, then there are two alternatives:
(a) u(·, c) = 0 on ξ(c), ∞ ;
(b) u(·, c) < 0 on 1 + ξ(c), ∞ and u(x, c) → −∞ as x → ∞.
Lemma 3.5 Suppose that ξ(c) < ∞. Then, by definition,
u(ξ(c), c) = 0
and
u(x, c) > 0
for
− ∞ < x < ξ(c),
and we may distinguish two alternatives:
1. We have
u(x, c) 6
1
2
for
x ∈ [ξ(c) − 1, ξ(c)].
In that case:
u(x, c) = 0
for every
x > ξ(c).
2. We have
u(x, c) >
1
2
for some
x ∈ [ξ(c) − 1, ξ(c)].
Then u(x, c) is nonincreasing for every x > ξ(c).
Moreover, if u(ξ(c)−1, c) < 12 , then u(x, c) is strictly decreasing for x > ξ.
Define
def
A(c) = −u 1 + ξ(c), c
Then A(c) > 0 and for x > 1 + ξ(c) we have
u(x, c) = −A(c) exp
In particular, limx→∞ u(x, c) = −∞.
95
x − 1 − ξ(c) c
.
(3.69)
Proof. We write the proof in terms of the variables t = x − 1 and y = 1 − u,
and we put τ = ξ − 1.
By assumption, τ < ∞ and by the definition of ξ, we have y < 1 on (−∞, τ )
and y(τ ) = 1. The equation now becomes
( 0
cy (t) = y(t) − 1
if t > 0 and y(t − 1) > 21 ,
(3.70a)
cy 0 (t) = y(t) − 2y(t − 1)
t > 0 and y(t − 1) 6 21 ,
if
(3.70b)
so that
cy 0 (t) > y(t) − 1
for
t > τ.
(3.71)
Remembering that y(τ ) = 1, we conclude that y(t) > 1 for all t > τ .
In Case (2a) we have y > 21 on (τ − 1, τ ), so that we have equality in (3.71),
and
y(t) = 1
for all
t > τ.
(3.72)
On the other hand, in Case (2b) we have y <
Thus, if we write
1
2
on some subinterval of (τ −1, τ ).
cy 0 (t) = y(t) − 2y(t − 1) = y(t) − 1 + f (t),
where
f (t) = 1 − 2y(t − 1),
we see that f (t) > 0 and f (t) > 0 on some subset of (τ − 1, τ ). Therefore
y(τ + 1) > 1.
For t > 1 + τ (c), we know that y(t) obeys the ODE (3.70a). Solving it with
the initial condition y(1 + τ (c), c) = 1 + A(c) gives us the expression (3.69).
In the following Proposition we summarise the results of a numerical investigation, identifying the values of the speed c with the three cases (a), (b) and
(c) enumerated above.
Proposition 3.2 There exists a critical speed cunb ∈ (0, cbif) (cunb ≈ 1.455)
such that:
1. For c > cunb : u(x, c) > 0 for every x > 1.
2. For c = cunb there exists a point x̃ > 1 (x̃ ≈ 5.449) such that u(x, cunb ) > 0
for x < x̃ and u(x, cunb ) = 0 for x > x̃.
96
3. For 0 < c < cunb there exists a point xc < x̃ such that u(xc , c) = 21
and u(1 + xc , c) < 0. By Proposition 3.5, this implies that there exists a
constant B(c) > 0 such that
u(x, c) = −B(c) exp
x − (x + 1) c
c
for
x > 1 + xc .
In particular, limx→∞ u(x, c) = −∞.
The range cunb < c 6 cbif . We now focus on the interval (cunb , cbif ]. So far
we can only say that u(x, c) remains positive for all x ∈ . In the following we
show that under certain conditions, which we can verify numerically, for some
values of c in this interval, there exists a point xper (c) such that the orbit u(x, c)
is periodic for x > xper (c).
We introduce the points
x1 (c)
x2 (c)
1
sup x > 1 : u(·, c) > on (1, x) ,
2
1
sup x > x1 (c) : u(·, c) < on (x1 (c), x) ,
2
def
=
def
=
(3.73)
(3.74)
whenever they exist, and the constant
def
L(c) = u 1 + x1 (c), c .
The points x1 (c) and x2 (c) are the first two positive values of x where u(x, c)
crosses the line u = 12 . On the basis of numerical evidence – see also Conjecture
8.1, – the point x1 (c) exists if c 6 c1 and x2 exists if c < c1 .
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-5
0
5
10
15
20
-5
(a) c = 1.6
0
5
10
(b) c = 1.5
97
15
20
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-5
0
5
10
15
20
-5
(c) c = 1.46
0
5
10
15
20
(d) c = 1.456
Figure 3.5: Eventually periodic orbits for different values of c.
In the following Proposition, we prove the existence of the announced eventually periodic solutions under the condition that
x2 (c) − x1 (c) > 1.
(3.75)
Note that this condition implies that L(c) 6 21 . We characterize numerically the
values of c ∈ (cunb , cbif ) for which this condition is satisfied, and in Figure 3.5
we show some eventually periodic orbits for different values of c. Note that the
amplitude and the period both increase as c decreases. Some of those properties
are described in Theorem 3.8.
Theorem 3.8 Let u(x, c) be the solution of Problem (8.1) and let c be such that
(a) The points x1 (c) and x2 (c) exist.
(b) The distance between x1 (c) and x2 (c) is large enough, i.e. (3.75) is satisfied.
Then there exists a point xper (c) 6 0 such that
u(x, c) =
1 x/c
e
2
for
x 6 xper
and u(x, c) is periodic for x > xper (c). The period T (c) is given by
1 T (c) = 1 + x1 (c) + c log
.
2L(c)
(3.76)
Proof. We write ti = xi − 1 (i = 1, 2). Then, by assumption (3.75),
t2 (c) − t1 (c) > 1
98
(3.77)
Moreover, writing y(t) = 1 − u(t), we obtain
y(1 + t1 (c), c) = 1 − u(1 + x1 (c), c) = 1 − L.
Since y(t) >
problem
1
2
(3.78)
for t1 6 t 6 t2 , we can find y(t) on (t1 + 1, t2 + 1) by solving the
(
cy 0 (t) = y(t) − 1
y(t2 ) = 21 .
for t1 + 1 < t < t2 + 1,
(3.79)
Note that because of assumption (3.75)
t1 + 1 6 t2 < t2 + 1.
The solution of Problem (8.14) is given by
y(t) = 1 −
t − t 1
2
exp
2
c
for t1 + 1 6 t 6 t2 + 1.
Since y(t1 + 1) = 1 − L by (3.78), we obtain the following relation between t1 ,
t2 and L:
1 .
(3.80)
t2 = 1 + t1 + c log
2L
Now, consider the following auxiliary function:
z(t) = y(t + t2 + 1)
for
t > −1.
(3.81)
Then z(t) verifies the same delay-differential equation (3.70b) as y(t), and
z(t) = y(t)
for
− 1 6 t 6 0.
Hence, by uniqueness this implies that
z(t) = y(t + t2 + 1) = y(t).
In other words, y(t) is periodic and the period is given by T = t2 + 1. From
(3.80) we obtain that the period is given as in (3.76).
To complete the proof, we need to determine tper (c). This is found to be the
value which satisfies y(tper ) = 1 − L. A simple computation shows that tper is
given by
1 tper (c) = −1 − c log
< −1.
(3.82)
2L(c)
99
Remark. Note that the condition (3.75) is essential. If it is not satisfied, we
can still conclude is that z(t) and y(t) verify the same DDE (3.70b). However,
the initial condition is different. Indeed, in this case what we know is that
there exists an interval of length t2 − t1 < 1 on which y(t) and z(t) coincide.
Specifically, z = y on [0, t2 − t1 ] , which is not enough for uniqueness.
To finish this section, it remains to characterize numerically for which values
of c condition (3.75) is satisfied and hence can apply Theorem 3.8.
Proposition 3.3 There exists cunb < cper < cbif (cper ≈ 1.618) such that condition (3.75) is satisfied if cunb < c 6 cper . By Theorem 3.8, this implies that
there exists a point xper (c) 6 0 given by (3.82) and a constant T (c) > 0 given
by (3.76) such that
u(x, c) = u(x + T (c), c)
for every
x > xper (c).
Moreover,
lim T (c) = 1 + x1 (cper ),
(3.83)
lim xper (c) = 0.
(3.84)
lim T (c) = ∞,
c↑cper
lim xper (c) = −∞,
c↑cper
c↓cunb
c↓cunb
def
Discussion. In the interval I = (cunb , cbif ) the value x1 (c) given by (3.73) can
be expected to be well defined and finite. This can be seen in two ways: by
using Part 3 of Conjecture 3.1 or by recalling that the spectrum of the operator
associated to the linear DDE (3.66b) contains eigenvalues with positive real
part. Hence, the amplitude of the oscillations around the uniform steady state
u = 1 increases and eventually the orbit crosses u = 21 .
On I the point x2 (c) defined by (3.74) is also well defined and finite. The
argument works as follows: the orbit crosses the line u = 21 at x = x1 and also
u > 0 on because c > cunb , by Proposition 3.2. Suppose that the orbit verifies
that 0 < u(·, c) < 21 on (x1 (c), ∞). Then, eventually it must satisfy the ODE
(3.66a). The ODE forces the orbit to cross the line u = 21 , which contradicts
our assumption. Hence, x2 (c) is well defined and finite.
Thus, we have that x1 (c) and x2 (c) are well defined on I. By using continuity
of solutions with respect to parameter c, we know that
lim x2 (c) − x1 (c) = ∞
c↓cunb
because x2 (c) tends to infinity. In particular, we can find a small right-neighbourhood
of cunb where condition (3.75) is satisfied. Specifically, there exists some ε > 0
100
such that condition (3.75) holds on (cunb , cunb + ε]. We define
def
cper = sup{c ∈ I : condition (3.75) holds on (cunb , c)}
(3.85)
This definition gives us the criterium to find cper : it must satisfy the equation
1
u 1 + x1 (c), c = .
2
(3.86)
The limits in (3.83) are obtained from (3.76) and (3.80) by using the fact that
lim L(c) = 0
c↓cunb
and
101
lim L(c) =
c↑cper
1
.
2
102
Chapter 4
The discrete
Swift-Hohenberg equation
4.1
Introduction
In the study of pattern formation the Swift-Hohenberg equation,
2
∂u
∂2
u − u3 ,
0 < x < L, t > 0,
= αu − 1 + 2
∂t
∂x
(SH)
is a classical model equation. Here, α and the length L of the domain are
eigenvalue parameters. This equation is known to possess a rich structure of
stationary solutions with complex graphs [59], many of them locally stable.
An important issue is here the selection of patterns that develop as the system
evolves to a final state. Since the Swift-Hohenberg equation is a gradient system
[20, 53, 54] final states will be stationary solutions, provided they are isolated.
This limits the choice to the set of equilibrium states and thus simplifies the
problem to a certain extent.
However, this is still a complicated problem, and in the literature several
simplified versions of the Swift-Hohenberg equation have been studied [48]. In
the present paper we introduce another simplification, reducing (SH) to an
equation on a finite lattice with N + 1 nodes. We establish some properties of
solutions x = (x0 , . . . , xN ), keeping N general, but our main results concern the
dynamics of the system when N = 4 and the outer nodes, x0 and x4 , are kept
fixed. This leaves a three-dimensional system of ordinary differential equations,
103
or, in the presence of symmetries, an even lower-dimensional system, which is
simple enough to understand in great detail, whilst still complex enough to show
several of the properties of the original SH equation, such as the selection of
different patterns as time runs to infinity. The objective of this paper is to use
this simple model to shed light on the dynamics of the SH equation.
It is instructive to place the study of the Swift-Hohenberg equation against
the backdrop of the Fisher-Kolmogorov equation
∂ 2u
∂u
=
+ u(1 − u2 ),
∂t
∂x2
0 < x < L, t > 0,
(FK)
also a gradient system, but of second order. The dynamics of the CauchyDirichlet problem for this equation was first studied in a classical study by
Chafee & Infante [12], who found that for L 6 π 2 , the origin is a global attractor,
whilst for any L > π 2 there exist two stable stationary solutions, ±v(x), where
v(x) > 0 for all x ∈ (0, L). All other stationary states, and as L increases, their
number also increases, are unstable. Therefore, for this equation the selection
of the final pattern is quite simple.
Thus, in this paper we consider two lattice equations, one emulating the
Swift-Hohenberg equation, and the other the Fisher-Kolmogorov equation. We
begin with the latter, which is simpler.
Fisher-Kolmogorov equation: We first rescale the variables, putting x =
Lx∗ , t = L2 t∗ . This transforms the domain to the interval (0, 1), and we obtain
ut = uxx + σu(1 − u2 ),
0 < x < 1,
t > 0,
σ = L2 ,
where we have dropped the asterisks again. We then replace the diffusion term
by symmetric coupling of nearest neighbours, as is done in finite difference
approximations, and obtain the equation
x0n (t) = ∆xn (t) + σ f (xn (t)),
n = 1, . . . , N − 1,
(4.1)
in which ∆ denotes the discrete Laplacian operator:
∆xn = xn+1 − 2xn + xn−1 .
(4.2)
Swift-Hohenberg equation: As before, we rescale the variables and put
x = Lx∗ . This yields the equation
ut = −γ 2 uxxxx − 2γuxx − (1 − α)u − u3 ,
104
0 < x < 1, t > 0,
γ=
1
.
L2
We replace the second order derivative by the symmetric coupling term involving nearest neighbours (4.2), and the fourth order derivative by a symmetric
coupling term involving the nearest– as well as the next-nearest neighbours.
This results in the equation
u0n (t) = −γ 2 ∆2 un (t) − 2γ∆un (t) − (1 − α) un (t) − u3n (t),
(4.3)
in which n = 1, . . . , N − 1, and
∆2 un = un+2 − 4un+1 + 6un − 4un−1 + un−2 .
(4.4)
Note that the discrete Fisher-Kolmogorov equation (4.1) has one parameter,
σ = L2 , whilst the discrete Swift-Hohenberg equation (4.3) has two parameters,
α > 0 and γ = 1/L2. As was shown in [59], the number of stationary states,
and the complexity of their graphs, increases as α and L increase, and hence as
γ decreases.
Like the Swift-Hohenberg and the Fisher-Kolmogorov equation, both discrete equations are gradient systems. In the next section we define the corresponding actions JF and JS .
In a recent paper [53] (see also [54]) numerical simulations were made of the
evolution of solution profiles of the Swift-Hohenberg equation and of the pattern
selection for different values of the length L of the interval. At critical values
of L, the final pattern was found to switch abruptly from one type to another,
with a different number of nodes. In this paper, by studying a much simplified
system, we wish to shed light on this switching phenomenon and to understand
the dynamics that leads to it.
Since both discrete equations are gradient systems, limiting profiles are all
stationary solutions, because they are isolated. Thus, we begin with an analysis
of the equilibrium profiles. This leads to nonlinear eigenvalue problems, the
eigenvalue being L for the FK-equation and γ for the SH-equation, when we
keep α fixed. In Figure 4.1, we draw the norm of the symmetric equilibria
versus the length L, first for the discrete FK equation and then for the discrete
SH equation, with fixed α = 0.7. The stable equilibria are shown with solid line
and the unstable ones with dashed lines.
4.2
Preliminaries
We begin with some general properties of the discrete versions of the second
order Fisher-Kolmogorov equation and the fourth order Swift-Hohenberg equation.
105
2
1.5
1.5
||u||
||u||
1
1
0.5
0.5
0
0
1
2
3
0
0
4
1
2
L
3
4
5
L
(a) Discrete FK equation.
(b) Discrete SH equation with
α = 0.7.
Figure 4.1: Graph of kuk versus L.
The discrete Fisher-Kolmogorov equation:
 0
n = 1, 2, . . . , N − 1,

 un = ∆un + σf (un )
un = 0
n = 0, N, t > 0,


un (0) = un,0 ,
n = 1, 2, . . . , N − 1,
t > 0,
(4.5a)
(4.5b)
(4.5c)
Here, ∆ denotes the discrete Laplacian operator defined in (4.2), the parameter
σ is positive, and the function f is given by
f (s) = s − s3 .
(4.6)
Let us introduce some notation. We will denote vectors in
given a function f defined from to , we will write
f (u) = f (u1 ), f (u2 ), . . . , f (uN −1 )
With this notation equation (4.5a) can be rewritten as
u0 (t) = −Au(t) + σf (u),
T
N −1
by u, and
.
(4.7)
where A = (aij ) is the symmetric positive definite matrix with ajj = 2 and
aj,j+1 = −1, aj+1,j = −1, and all other entries equal to 0. Its eigenvalues λj are
106
given by 2 1 − cos( πj
N ) for j = 1, . . . , N − 1. Note that for σ = 0, the system
(4.5) is linear and that the solution is given by
u(t) = e−At u(0),
(4.8)
where u(0) = (u1,0 , . . . , uN −1,0 )T . Since λj > 0 for j = 1, . . . , N − 1, the origin
is asymptotically stable.
Associated with equation (4.5) we define the Action
def
JF (u) = J1 (u) − σ
in which
def
J1 (u) =
N
−1
X
F (un ),
(4.9)
n=1
N
1 X
1 T
(un − un−1 )2 ,
u Au =
2
2 n=1
where u0 = 0 and uN = 0, and F is a primitive of f :
Z s
s2
s4
def
f (t) dt =
− .
F (s) =
2
4
0
(4.10)
(4.11)
We can write equation (4.5a) as a gradient system involving the action J F :
u0 (t) = −∇JF u(t) .
(4.12)
The discrete Swift-Hohenberg equation:
 0
2 2

 un = −γ ∆ un − 2γ∆un − fα (un ), n = 1, . . . N − 1, t > 0,
un = 0,
∆un = 0,
n = 0, N,
t>0


un (0) = un,0 ,
n = 1, . . . N − 1.
(4.13a)
(4.13b)
(4.13c)
The operator ∆ is the discrete Laplacian operator, defined in (4.2), ∆2 is the
discrete bi-Laplacian defined in (4.4) and we introduce virtual nodes at n = −1
and n = N + 1, where we define u−1 = −u1 and uN +1 = −uN −1 , in order to
make ∆ and ∆2 well defined at all the nodes, and ensure that u is odd with
respect to the end points. The function fα is given by
fα (s) = s3 + (1 − α)s,
α > 0.
(4.14)
Equation (4.13a) can be rewritten as
u0 (t) = −γ 2 A2 u(t) + 2γAu(t) − fα u(t) ,
107
(4.15)
where the matrix A is as in (4.7). In general we will focus on α ∈ (0, 1) and
γ > 0, although we will sometimes give results for α > 1.
Associated with equation (4.13) we define the action
def
JS (u) = γ 2 J2 (u) − 2γJ1 (u) +
3
X
Fα (un ),
(4.16)
n=1
where Fα denotes the potential
Z s
1
1−α 2
Fα (s) =
fα (t) dt = s4 +
s ,
4
2
0
(4.17)
which has a single well if α < 1 and a double well if α > 1. The functionals J1
(cf. (4.10)) and J2 are defined as follows:
1
J1 (u) = uT Au,
2
N
−1
X
1 T 2
J2 (u) = u A u =
(∆un )2 .
2
n=1
def
(4.18)
As in (4.12), we can write equation (4.13a) as a gradient system:
u0 (t) = −∇JS u(t) .
It follows that the critical points of JF , respectively JS , are precisely the equilibrium states of Problems (4.5) and (4.13).
Thus, Problems (4.5) and (4.13) are gradient systems, and it is readily shown
that both actions, JF and JS , are coercive, bounded below, and that they decrease along orbits. For both systems, finding equilibrium states can be reduced
to a problem of searching for roots of a certain polynomial, a procedure which
will be used in later sections of this paper. In particular, this implies that
the set of equilibrium states is finite and discrete. Therefore, we may conclude
[34] that every orbit tends to an equilibrium state as t → ∞, i.e. for every
u0 ∈ N −1 there exists an equilibrium state u∞ = u∞ (u0 ) such that the orbit
u(t) = u(t, u0 ), which starts at u0 , tends to u∞ :
lim u(t) = u∞ .
t→∞
Thus, in the description of the large time behaviour of the discrete FisherKolmogorov and the discrete Swift-Hohenberg equation, equilibrium solutions
play a pivotal role. We therefore first focus on giving a complete characterisation
of the sets of equilibrium states of these equations.
108
We conclude this section with a few observations about symmetry properties
of solutions. We consider the following linear operators in N −1 :
L1 (u1 , u2 , . . . , uN −2 , uN −1 ) = (−u1 , −u2 , . . . , −uN −2 , −uN −1 ),
L2 (u1 , u2 , . . . , uN −2 , uN −1 ) = ( uN −1 , uN −2 , . . . , u2 , u1 ),
L3 (u1 , u2 , . . . , uN −2 , uN −1 ) = (−uN −1 , −uN −2 , . . . , −u2 , −u1 ).
(4.19)
Note that L3 = L1 L2 = L2 L1 . The unique fixed point of L1 is the origin,
while the fixed points of the other operators are given by
Π2
def
Π3
def
=
=
u∈
u∈
N −1
N −1
for every j = 1, . . . , N − 1 .
for every j = 1, . . . , N − 1 .
: uj = uN −j
: uj = −uN −j
(4.20)
The operator L1 maps each point u onto the symmetric point with respect to
the origin, L2 maps each point u onto the symmetric point with respect to the
linear subspace Π2 . Finally, L3 maps each point u onto the symmetric point
with respect to the linear subspace Π3 .
Invariance of the flow with respect to Π2 in the discrete model corresponds
to the following property for every t > 0 in the continuous model.
u(x) = u(L − x),
for every x ∈ (0, L).
(4.21)
Similarly, invariance of the flow with respect to Π3 in the discrete model corresponds to the following property holding for every t > 0 in the continuous
model.
u(x) = −u(L − x),
for every x ∈ (0, L).
(4.22)
In the following lemma we show that the flow φt (·) associated with either
Problem (4.5) or Problem (4.13) commutes with each of the operators Li .
Lemma 4.1 The flows associated with Problem (4.5) and Problem (4.13) have
the properties
φt Li (u) = Li φt (u)
for every t > 0 and i = 1, 2, 3.
(4.23)
The proof is readily given by inspection.
Points contained in Π2 will be called symmetric and points contained in Π3 will
be called antisymmetric.
Corollary 4.1 The sets Π2 and Π3 are invariant.
109
Proof. Let u0 ∈ Π2 . Then L2 (u0 ) = u0 and by Lemma 4.1, L2 φt (u0 ) =
φt (u0 ). Therefore, φt (u0 ) is a fixed point of L2 and hence, φt (u0 ) ∈ Π2 . The
proof of the invariance of Π3 is similar. This corollary allows us to call solutions in Π2 symmetric and solutions in Π3
antisymmetric.
We will show in the following sections that for σ 6 2 and for α 6 (2γ − 1)2 ,
the set Π2 attracts solutions of respectively the Fisher-Kolmogorov equation
and the Swift-Hohenberg equation, and that u∞ ∈ Π2 .
4.3
The discrete Fisher-Kolmogorov equation
In this section we investigate the discrete Fisher-Kolmogorov equation in a lattice consisting of 5 nodes, u0 , . . . u4 , of which the end nodes u0 and u4 are fixed
and assumed to be equal to zero. For the intermediate nodes, u1 , u2 and u3 we
then find the following system of equations
 0
(4.24a)

 u1 = −2u1 + u2 + σf (u1 ),
0
u2 = u1 − 2u2 + u3 + σf (u2 ),
(4.24b)

 0
u3 = u2 − 2u3 + σf (u3 ).
(4.24c)
We will be studying the evolution of the solution u(t, u0 ) which starts at u0 ∈
3
. As we saw in Section 2, this solution exists for all time and converges to an
equilibrium point u∞ (u0 ) ∈ 3 .
We first derive a few general properties of solutions of Problem (4.24), and
then focus on symmetric solutions, i.e. solutions in Π2 , in Section 3.1, on
antisymmetric solutions in Section 3.2 and then on nonsymmetric solutions in
Section 3.3.
We define the ellipse
def (4.25)
Sρ = (u1 , u2 , u3 ) ∈ 3 : u21 + u1 u3 + u23 = ρ ,
and we show:
Lemma 4.2 Let u(t) be a solution of Problem (4.24). Then
u(t) → Π2
u(t) → Π2 ∪ Sρ
as
as
t→∞
t→∞
where ρ = 1 − σ2 .
110
if
0<σ62
if
σ > 2,
Proof. We assume that u(t) is a nontrivial solution, and that u(t) tends to a
limit u∞ 6= 0 as t → ∞. Put
def
z = u1 − u3 .
Then, subtracting the equation for u3 from the one for u1 we obtain
z 0 = −(2 − σ) − σ u21 + u1 u3 + u23 z.
(4.26)
(4.27)
Suppose that z(0) > 0. Then, equation (4.27) implies that z(t) > 0 for all t > 0.
Since
u21 + u1 u3 + u23 = 21 (u1 + u3 )2 + 12 (u21 + u23 ) > 0,
because u 6= 0, it then follows that
z 0 < −(2 − σ)z,
and hence that
z(t) < z(0) e−(2−σ)t .
(4.28)
Similarly, if z(0) < 0. Hence, if σ ∈ [0, 2), then z(t) → 0 as t → ∞, and
u(t) → Π2 .
Next, suppose that σ = 2. Then equation (4.27) becomes
z 0 = −2z (u21 + u1 u3 + u23 ).
Since we assume that u(t) tends to a nontrivial limit as t → ∞, it follows that
u21 + u1 u3 + u23 tends to a positive limit and hence we may conclude again that
|z(t)| → 0 exponentially.
Finally, suppose that σ > 2. Because u(t) tends to a nonzero limit as t → ∞,
it follows that the limit of the right hand side of (4.27) exists, and hence that
z 0 (t) → 0. This means that u∞ ∈ Sρ ∪ Π2 .
4.3.1
Symmetric solutions
When u1 = u3 the system (4.24) reduces to
(
u01 = −2u1 + u2 + σf (u1 ),
u02
= 2u1 − 2u2 + σf (u2 ).
(4.29a)
(4.29b)
Before giving a complete characterisation of the equilibrium states, let us first
give a brief description of the isoclines, which will help us to understand the
111
structure. In Figure 4.2 we have plotted the different scenarios for different
values of σ. When σ is close to zero we only have the trivial equilibrium state.
Then, when σ is increased,
two new equilibria bifurcate (supercritically) from
√
the origin at σ = 2 − 2, one into the first quadrant, and one into the third
quadrant of the (u1 , u2 )-plane. When σ exceeds the value
√ 2, the slope of the
isoclines at the origin becomes negative and at σ = 2 + 2 two new equilibrium
states bifurcate (supercritically) from the origin, this time into the second and
the fourth quadrant. Thus, on the branch emanating from the first bifurcation
point u1 and u2 have the same sign, whilst on the branch emanating from
the second
√ bifurcation point u1 and u2 have opposite sign. Finally, at a value
σ > 2 + 2 (σ̂ = 2 + µ̂ ≈ 6.85) four new equilibria appear from saddle-node
bifurcations. The value µ̂ is found to be the unique positive root of a sixth order
polynomial.
4x6 − 123x4 + 768x2 − 2048 = 0
(µ̂ ≈ 4.85).
In Figure 4.3, we show these results in a bifurcation diagram. We represent the
stable equilibria with solid lines and the unstable ones with dashed lines.
In the next theorem we give the complete characterisation of the set of
symmetric equilibrium solutions of Problem (4.29), as well as their stability
properties.
Theorem 4.1 The number of symmetric equilibria of equation (4.5) is given
by:
√
1. If 0 < σ < 2 − 2, then the origin is the unique equilibrium. It is stable.
√
√
2. If 2 − 2 < σ < 2 + 2, then there exist two nontrivial
symmetric equilib√
ria. They bifurcate from the origin at σ = 2 − 2 (supercritical pitchfork
bifurcation) into the first and the third quadrant
and they extend to σ =, ∞.
√
They are stable nodes for every σ > 2 − 2 and in this regime, the origin
is a saddle.
√
3. If 2 + 2 < σ < σ̂, then there exist four nontrivial equilibrium states.
√ The
two new symmetric equilibria bifurcate from the origin at σ = 2 + 2 (supercritical pitchfork bifurcation) into the second and the fourth quadrant,
√
and they extend to σ = ∞. They are saddles for every σ > 2 + 2 and
the origin becomes an unstable node.
4. If σ > σ̂, then there exist eight nontrivial equilibria. The four new symmetric equilibria emerge at σ = 2 + µ̂ from a saddle-node bifurcation at
112
1
1
0.8
0.6
0.4
0.5
PSfrag replacements
γ
α
γ
σ
σ
1
u1
u2
u2
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.5
0
u1
(a) 0 < σ < 2 −
0.5
−1
−1
√
2
−0.5
(b) 2 −
0
u1
0.5
1
√
√
2<σ <2+ 2
2
1
0
−1
−2
−2
−1.5
−1
−0.5
(c) 2 +
0
u1
0.5
1
PSfrag replacements
γ
α
γ
σ
σ
1.5
2
u1
√
2 < σ < σ̂
1
u2
g replacements
γ
α
γ
σ
σ
u1
0
−0.5
2
u2
replacements
γ
α
γ
σ
σ
u1
0
−1
−2
−2
−1
0
u1
1
2
(d) σ > σ̂
Figure 4.2: Isoclines for different values of σ.
a point in the second, respectively the fourth, quadrant. The equilibrium
states with the highest value of |u2 | are stable nodes and the other two are
saddles.
The proof involves a detailed study of the phase plane, and of the two isoclines
in particular. We give the details in the Appendix.
From Theorem 4.1 and inspection of the phase plane we can draw the following conclusions:
√
√
1. For σ 6 2 − 2 the trivial solution u = 0 is stable and for all σ > 2 − 2
it is unstable.
113
2
PSfrag replacements
γ
α
γ
σ
σ
u1
||u||
1.5
1
0.5
0
0
5
10
15
20
σ
Figure 4.3: Bifurcation diagram of the symmetric equilibria
√
2. For any σ > 2 − 2 there exists a unique symmetric equilibrium state
u∗ in the first quadrant and its domain of attraction includes the first
quadrant. Similarly, −u∗ is the unique symmetric equilibrium state in the
third quadrant, attracting all symmetric orbits that start in this quadrant
as t → ∞.
For symmetric solutions the action JF (u) defined in (4.9) is given by
JF (u) = u21 + (u2 − u1 )2 −
σ
σ
(2u21 + u22 ) + (2u41 + u42 ).
2
4
(4.30)
√
3. For σ 6 2 −√ 2 the trivial solution is the global minimiser of JF (u), and
for σ > 2 − 2 the global minimiser is nontrivial.
Since JF decreases along orbits, it follows from the second observation that JF
attains its minimum value in the first quadrant at u∗ . In the following theorem
we show that
JF (u∗ ) = min{JF (y) : y ∈ 2 }.
114
Theorem 4.2 Let σ > 2 −
JF in 2 .
√
2. Then u∗ and −u∗ are the global minimisers of
Proof. We have seen in Theorem 4.1 that because u∗ attracts all the orbits that
start in the first quadrant, it is the global minimiser of JF in this quadrant.
Suppose that JF has a local minimum at a point y∗ in the interior of the fourth
quadrant. Write
u1 = y 1
and
u2 = −y2 .
Then the point u = (u1 , u2 ) lies in the first quadrant. Moreover, we see from
(4.30) that
σ
σ
(2y12 + y22 ) + (2y14 + y24 )
2
4
= JF (y) + (y2 + y1 )2 − (y2 − y1 )2
= JF (y) + 4y1 y2 < JF (y).
JF (u) = y12 + (y2 + y1 )2 −
Since JF (u) > JF (u∗ ) for all u in the first quadrant, it follows that JF (y) >
JF (u∗ ), so that y cannot be a global minimiser in 2 .
So far these properties all mirror properties of the Fisher-Kolmogorov equation. The next property, however, introduces a difference between the discrete
and the continuous problem.
4. For every σ > σ̂, there exist stable states ξ and −ξ in the second, respectively
the fourth quadrant. Therefore, there exist initial data which yield orbits that
converge to a final state which does not have one sign.
In conclusion we note that the problem discussed in this subsection has
much in common with linear diffusion in a slab under the influence of nonlinear
boundary conditions, as was discussed by Aris [5] and Aronson & Peletier [6].
4.3.2
Antisymmetric solutions
When u3 = −u1 and u2 = 0 the system (4.24) reduces to
u01 = (σ − 2) u1 − σu31 .
(4.31)
We will call equation (4.31) the reduced antisymmetric system. The following
Theorem gives us the main result about existence and stability of nontrivial
antisymmetric solutions. Recall that nontrivial antisymmetric points are of the
form u = k(1 0 − 1) for some k ∈ , k 6= 0.
115
Theorem 4.3 Let u(t) be a nontrivial antisymmetric solution.
1. If σ 6 2, then
u(t) → 0,
as
t → ∞.
2. If σ > 2, then
u(t) → ±
r
1−
2
(1 0 − 1) ,
σ
as
t → ∞,
where sgn (u1 (t)) = sgn(k).
With respect to the system (4.24), the nontrivial antisymmetric equilibria
±u(σ) are locally unstable for every σ > 2.
Proof. Existence of nontrivial antisymmetric equilibria follows directly from
equation (4.31). When σ 6 2, the antisymmetric solution converges to the
origin and for σ > 2, the origin loses stability and the bifurcating nontrivial
antisymmetric equilibria are stable. As equation (4.31) is one dimensional, the
final profile is determined by the sign of u1 (0).
Let us study the question of stability of the nontrivial equilibria for σ > 2 in
the system (4.24). We linearise the system around the nontrivial antisymmetric
equilibrium. The stability matrix is given by

4 − 2σ

1
0

1
0
.
σ−2
1
1
4 − 2σ
and its eigenvalues are the following:
λ1 (σ)
λ2 (σ)
λ3 (σ)
= 4 − 2σ < 0,
√
9σ 2 − 36σ + 44
σ
< 0,
= 1− −
2
2
√
9σ 2 − 36σ + 44
σ
= 1− +
> 0,
2
2
so that the equilibrium is unstable.
116
4.3.3
Nonsymmetric equilibria
Since by Lemma 4.25, any solution of the full system tends to Π2 as t → ∞ if
σ 6 2, we have the following nonexistence theorem.
Theorem 4.4 For σ 6 2 there exist no nonsymmetric equilibrium states.
The result of this theorem is optimal because for every σ > 2 we have two
antisymmetric equilibrium states bifurcating from the trivial solution at σ = 2,
found in the previous subsection.
The result of a numerical search for more nonsymmetric equilibria is summarized in the following proposition, and the bifurcation picture is shown in
Figure 4.4, where we plot kuk vs. σ.
Proposition
4.1 For σ ∈ (0, 2], every equilibrium is symmetric. For σ ∈
√
(2, 2 +
2
2),
every
equilibrium is either symmetric or antisymmetric. For σ >
√
2 + 2 2, there exist 16 new nonsymmetric√equilibrium states. They all emerge
from saddle-node bifurcations at σ = 2 + 2 2.
2
PSfrag replacements
γ
α
γ
σ
σ
u1
||u||
1.5
1
0.5
0
0
5
10
15
20
σ
Figure 4.4: Bifurcation diagram of the nonsymmetric equilibrium states
Note that if u is a nonsymmetric equilibrium state, then so are −u and
±L2 (u), where L2 denotes the operator that reflects u with respect to the
117
central node u2 . Thus, the 16 new nonsymmetric solutions consist of 4 different
branches of solutions.
4.4
The discrete Swift-Hohenberg equation
Here we turn to the discrete Swift-Hohenberg equation on a lattice of 5 nodes,
u0 , . . . , u4 . At the end points of the lattice we impose the conditions
u0 = 0,
u4 = 0
and
u−1 = −u1 ,
u5 = −u3 ,
where the virtual nodes u−1 and u5 need to be introduced to make the equation
for the points u1 , u2 and u3 well defined. We then obtain the following system
 0
2
2
2

 u1 = (−5γ + 4γ)u1 + (4γ − 2γ)u2 − γ u3 − fα (u1 ),
u02 = (4γ 2 − 2γ)u1 + (−6γ 2 + 4γ)u2 + (4γ 2 − 2γ)u3 − fα (u2 ),
(4.32)

 0
2
2
2
u3 = −γ u1 + (4γ − 2γ)u2 + (−5γ + 4γ)u3 − fα (u3 ),
where
fα (s) = (1 − α)s + s3 .
As with the discrete Fisher-Kolmogorov equation, for α sufficiently small, orbits
of Problem (4.32) eventually approach a symmetric state, i.e. they tend to the
plane Π2 . Specifically we have:
Lemma 4.3 Let u(t) be a solution of Problem (4.32). Then
u(t) → Π2
u(t) → Π2 ∪ Sδ
as
as
t→∞
t→∞
if
if
0 < α 6 (2γ − 1)2 ,
α > (2γ − 1)2 ,
where δ = α − (2γ − 1)2 .
The proof is similar to that of Lemma 4.2, and we omit it here.
In order to determine the linear stability of u = 0, we substitute u(t) = e−λt e
into the system (4.32) and omit higher order terms. This leads to an eigenvalue
problem with eigenvectors
√
√
e1 = (1, 2, 1)T ,
e2 = (1, 0, −1)T ,
e3 = (1, − 2, 1)T .
(4.33)
with corresponding eigenvalues
√
λ1 = −p − q 2,
λ2 = (2γ − 1)2 − α,
118
√
λ3 = −p + q 2,
(4.34)
where
def
def
p(α, γ) = −6γ 2 + 4γ − 1 + α,
Note that we can write λ1 and λ3 as
√
λ1 = {(2 − 2)γ − 1}2 − α
and
q(γ) = 4γ 2 − 2γ.
λ3 = {(2 +
√
2)γ − 1}2 − α,
(4.35)
(4.36)
Plainly, e1 and e3 together span the set of symmetric solutions, whilst e2
spans the set of anti-symmetric solutions. In fact, writing u(t) = φ(t)e2 , we
find the following equation for φ(t):
φ0 = {α − (2γ − 1)2 }φ − φ3 .
(4.37)
which yields the following result:
Theorem 4.5 Let δ = α − (2γ − 1)2 , and let u(t) be a nontrivial antisymmetric
solution such that u(0) = ke2 for some k ∈ .
(a) If δ 6 0, then
u(t) → 0
as
t → ∞.
(b) If δ > 0, then
u(t) →
4.4.1
√
δ · sgn(k) e2
as
t → ∞.
Symmetric dynamics
For symmetric solutions we have u1 = u3 , so that the system (4.32) reduces to
( 0
u1 = (−6γ 2 + 4γ)u1 + (4γ 2 − 2γ)u2 − (1 − α)u1 − u31 ,
(4.38)
u02 = (8γ 2 − 4γ)u1 + (−6γ 2 + 4γ)u2 − (1 − α)u2 − u32 ,
or, written more compactly,
(
u01 = pu1 + qu2 − u31 ,
u02 = 2qu1 + pu2 − u32 .
(4.39)
The two equations are coupled through the term involving the coefficient q.
Hence, if q = 0, which is the case when γ = 12 , the system uncouples and we
obtain two identical equations for u1 and u2 :

1
0
3


 u1 = α − 2 u1 − u 1 ,
(4.40)

1

0
3
 u2 = α −
u2 − u 2 .
2
119
We see that if α < 12 , then the origin is stable, and all orbits converge to it, and
if α > 12 it is unstable, and we have 8 nontrivial equilibrium states:
(±ρ, 0),
where ρ =
stable.
q
(0, ±ρ),
and
(±ρ, ±ρ),
α − 21 . The ones on the axes are unstable; the other ones are
The linear stability of the trivial solution u = 0 depends on the sign of the
eigenvalues λ1 and λ3 . If they are both positive the origin is stable; if one is
negative it is unstable. It follows from (4.39) that
λ1 < 0
if
where
γ1± (α)
√ !
√ 2
1+
1± α .
2
=
Similarly, we find that
λ3 < 0
if
where
γ3± (α)
Note that if α <
1
2
=
def
γ ∈ I1 = (γ1− , γ1+ ),
(4.41)
def
γ ∈ I3 = (γ3− , γ3+ ),
√ !
√ 2
1−
1± α .
2
(4.42)
then the intervals I1 (α) and I3 (α) are disjoint, that
γ1+ (α) →
1
2
and
γ3− (α) →
1
2
as
α→
1
,
2
and that they overlap if α ∈ ( 12 , 1).
When I1 and I3 are disjoint, the situation is relatively simple, and we discuss
this case first.
The case 0 < α <
1
2
As in Section 3 we first give an overview of the situation by following the
isoclines in the phase plane as γ decreases. Remembering that γ was introduced
as L−2 , this corresponds to the length L of the interval increasing and, as
expected from the continuous problem, the complexity of the solution set, and
120
the solution graphs, increases as well. In Figure 4.5 the phase plane is shown at
four different values of γ when α = 0.3.
We see that for γ > γ1+ the isoclines only intersect at the origin, so that the
origin is the unique equilibrium state. It is clearly a global attractor. However,
γ drops below γ1+ two nontrivial solutions ±u1 (γ) bifurcate from the origin, one
into the first quadrant, and one into the third. They continue to exist until γ
reaches γ1− , when they merge with the trivial solution again. For γ ∈ [γ3+ , γ1− ]
the origin remains the only equilibrium state, but at γ3+ two nontrivial solutions
u3 (γ) bifurcate
from the origin,PSfrag
this time
into the second and the fourth
frag replacements
PSfragagain
replacements
replacements
quadrant. They too, return to the trivial solution, at γ3− . For γ below γ3− the
γ
γ
γ
trivial solution is the unique equilibrium state. The situation is illustrated in a
α
α
α
bifurcation diagram in Figure 4.6. These results are summarised in the following
γ
γ
γ
theorem, in which
σ
σ
σ
σ
σ σ (α), γ + (α) .
I1 (α) = γ1− (α), γ1+ (α)
and
I3 (α) = γ3−
3
u1
u1
u1
γ
γ
γ
frag replacements
PSfrag replacements
PSfrag replacementsα
α
α
µγ
µγ
γµ
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
1
α
3
γ
1
α
3
γ
0.4
0.2
0
5
σ
9
σ
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
1
3
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
uα1
γ
α
µ
−1
−1
1
(a) γ = 10
1
0.8
0.6
1
3
0.4
0.2
0
5
5
−0.2
9
−0.8
α
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(b) γ = 1.5
1
0.8
0.4
−0.6
−0.4
−0.2
0
0.2
0.4
(d) γ = 0.49
0.6
0.8
1
0.2
α
0
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
(c) γ = 0.51
1
0.6
5
−0.2
0.4
0.2
0
−0.2
−0.4
9
−0.6
−1
−1
0.2
−0.2
1
3
−0.8
−0.8
0.4
0.8
0.6
−0.4
−0.6
−1
−1
−0.8
0
−0.4
9
0
5
σ
9
σ
−0.2
uα1
γ
α
µ
α1
γ3
σ5
σ9
uα
1
γ
α
µ
0.4
0.2
−0.6
−0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
(e) γ = 0.4
0.6
0.8
1
α
−1
−1
−0.8
−0.6
−0.4
−0.2
0
121
0.4
(f) γ = 0.1
Figure 4.5: Isoclines for α = 0.3 and different values of γ.
Theorem 4.6 Suppose that 0 < α < 21 .
0.2
0.6
0.8
1
(a) If γ ∈ + \ (I1 ∪ I3 ), then the origin is the unique equilibrium state, and it
attracts all orbits.
(b) If γ ∈ I1 , then there exist two equilibrium states ±u1 (γ), one in the first
quadrant, and one in the third. The origin is now unstable and ±u1 are
stable.
(c) If γ ∈ I3 , then there exist two equilibrium states ±u3 (γ), one in the second
in the fourth.
The origin is unstable and ±u3 are also
PSfrag replacements quadrant, and one PSfrag
replacements
γ unstable.
γ
α
α
The
γ proof of this theorem is similar to that
γ of Theorem 3.1, and we omit it.
σ
σ
σ
σ
u1
u1
1
0
0.9
−0.01
0.8
1
3
5
9
α
−0.02
α
µ
0.6
0.5
1
3
0.4
0.3
0.2
5
9
0.1
0
0
0.5
1
γ
1.5
2
2.5
α
3
(a) kuk vs. γ.
−0.03
Js
0.7
||u||
α
µ
−0.04
−0.05
−0.06
−0.07
0
0.5
1
γ
1.5
2
2.5
3
(b) Js vs. γ.
Figure 4.6: Graph of kuk and Js versus γ for α = 0.3
In Figure 4.6, we fix α = 0.3 and we plot the norm ku(γ)k versus γ, where
q
kuk = 2u21 + u22 ,
and the action J of all the equilibria versus γ. We can easily identify the intervals
I1 and I3 as those where there exist nontrivial symmetric stable equilibria and
the action J is negative.
The case 21 < α < 1
In this regime of α the intervals I1 and I3 overlap, and the structure of the solution set is much more complicated. In Figures 4.7 and 4.8 we show bifurcation
122
pictures for α = 0.7. In Figure 4.7 we have put the norm kuk along the vertical
axis. We represent stable equilibria with solid line and unstable equilibria with
dashed line. In Figure 4.8, we have put u1 and u2 along the vertical axis. Since
the situation near γ = 12 is rather complex, we also present a blow up of the
graph for u1 near γ = 12 .
α
µ
1.5
1
||u||
PSfrag replacements
γ
α
γ
σ
σ
u1
0.5
1
3
5
9
0
0
0.2
0.4
0.6
0.8
1
γ
α
Figure 4.7: Graph of kuk versus γ for α = 0.7
Again, we first inspect the evolution of the isoclines in the phase plane when
γ varies from large to small values. The first sequence of phase portraits is
shown in Figure 4.9. We see that as γ passes γ1+ , then as in the previous case,
two equilibrium solutions bifurcate from the trivial solution, one into the first
quadrant, and one into the third. We denote them again by ±u. Then, still as
before, at γ3+ a second pair of equilibrium states, ±y bifurcate from the trivial
solution, one into the second quadrant, and one into the fourth. Reducing γ
further, we encounter a saddle-node bifurcation at γ = γ∗+ , which creates two
new equilibrium states z∗ and z∗∗ in the second quadrant, and −z∗ and −z∗∗
in the fourth quadrant.
When we approach γ = 12 we have
u → (ρ, ρ),
y → (0, ρ),
z∗ → (ρ, −ρ),
123
z∗∗ → (ρ, 0),
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
3
PSfrag replacements
γ
α
γ
σ
σ
u1
γ 0.55
α 0.5
µ
1
0.5
0.45
0
1
3 0.4
−0.5
5
50.35
9
α
9
−1
0
1
2
α
3
(a) u1 and u2 vs. γ.
0.3
0.45
0.5
0.55
(b) u1 vs. γ, amplified.
Figure 4.8: Graph of u1 and u2 versus γ for α = 0.7
where ρ =
q
α−
1
2.
This is shown in Figure 4.10. Note that the axes are
1
2.
isoclines for γ =
For γ < 21 , the eight nontrivial equilibria gradually disappear by bifurcation
phenomena similar to the way they emerged. The first bifurcation is of saddlenode type at γ = γ∗− , and the disappearing equilibria are ±u and ±z∗∗ . As γ
passes through γ1− , the equilibria ±y disappear by a bifurcation to the origin.
Finally, reducing γ further, as γ passes through γ3− , the equilibria ±z∗ disappear,
again by a bifurcation to the origin. For γ < γ3− , the origin is the unique
equilibrium. This sequence of events is illustrated in the four plots of Figure 4.11
and summarised in the following theorem.
Theorem 4.7 Suppose that
1
2
< α < 1. Then, there exist
γ3− (α) < γ1− (α) < γ∗− (α) <
1
< γ∗+ (α) < γ3+ (α) < γ1+ (α),
2
such that:
(a) If γ ∈ + \(γ3− , γ1+ ), the origin is the unique equilibrium state and it attracts
all orbits.
(b) If γ∗− < γ < γ1+ , a pair of nontrivial equilibria ±u exist, one in the first
quadrant and one in the third. At γ = γ∗− , u has a saddle-node bifurcation
124
Sfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
u2
u2
0.5
0
1
3
replacements −0.5
5
−1
−1
PSfrag
−0.5
0
u1
0.5
γ9
α
γ
σ
σ
u1
γ
α
µ
1
(a) γ = 10
1
u
0.5
y
0
1
u
0.5
0
−1
−1
−u
−0.5
0
u1
−1
−u
−0.5
0
u1
0.5
1
α
(c) γ = 0.52
1
1
0.5
u
z*
y
z**
0
3 −0.5
5
9 −1
−y
0.5
(b) γ = 0.6
−z**
1
3 −0.5
5
9 −1
α
1
1
3
replacements −0.5
5
u2
γ9
α
γ
σ
σ
u1
γ
α
µ
u2
Sfrag
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
−1
−u
−0.5
−y
0
u1
−z*
0.5
1
(d) γ = 0.51
Figure 4.9: Isoclines for α = 0.7 and different values of γ.
and at γ1+ , it has a supercritical bifurcation from the origin. u is a stable
equilibrium and it is the minimizer for 21 < γ < γ1+ .
(c) If γ1− < γ < γ3+ , a pair of nontrivial equilibria ±y exist. For γ1− < γ < 12 ,
they are located in the second quadrant (respectively the fourth) and for
+
1
2 < γ < γ3 , they are located in the first quadrant (respectively, the third).
They are all saddles.
(d) If γ3− < γ < γ∗+ , a pair of nontrivial equilibria ±z∗ exist, one in the second
quadrant and one in the fourth. At γ = γ∗+ , it has a saddle-node bifurcation
and at γ = γ3− it has a supercritical bifurcation from the origin. ±z∗ are
125
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
0.5
z*
u
y
0
z**
3
−0.5
5
−u
−z**
1
−y
−z*
9
α
−1
−1
−0.5
0
0.5
1
Figure 4.10: Isoclines for α = 0.7 and γ = 0.5.
stable and they are minimizers for γ3− < γ < 21 .
(e) If γ∗− < γ < γ∗+ , a pair of nontrivial equilibria ±z∗∗ exist. For γ1− < γ < 12 ,
they are located in the second quadrant (respectively the fourth) and for
+
1
2 < γ < γ3 , they are located in the third quadrant (respectively, the first).
They are saddles.
The proof of this theorem is obtained analytically in a similar way as the
proof of Theorem 4.1 and we omit it here.
Next, we want to describe the bifurcation diagram of equilibria shown in Figure 4.8. A simple bifurcation theorem shows that around the critical values
γ1− , γ1+ , γ3− , γ3+ there exist local bifurcation branches. These branches can be
continued as long as the conditions for the Implicit Function Theorem hold,
that is, as long as the appropriate determinant does not vanish. Applying
this argument to this case, we find by a combination of analytical and numerical arguments that one branch can be extended smoothly along the interval
[γ1− , γ3+ ], and the other one can be extended continuously but not smoothly
along [γ3− , γ1+ ]. The reason for the lack of smoothness is that the corresponding
determinant mentioned above vanishes. This occurs precisely at γ∗− and γ∗+ :
they are the values of γ for which both isoclines touch tangentially. This is
summarised in the following proposition:
Proposition 4.2 There exists a smooth parametrization h1 : [0, 1] → [γ1− , γ3+ ]×
3
with components s → (γ, u1 , u2 , u3 ) and a continuous parametrization h2 :
[0, 1] → [γ3− , γ1+ ] × 3 such that
126
Sfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
z*
0
1
3
replacements −0.5
5
−1
−1
z**
−u
−0.5
−y
0
u1
0.5
γ9
α
γ
σ
σ
u1
γ
α
µ
1
(a) γ = 0.48
1
z*
0.5
0
1
*
z
y
0.5
0
−1
−1
−y
*
−z
−0.5
0
u1
0.5
1
0.5
1
(b) γ = 0.4
1
0.5
0
1
3 −0.5
5
9 −1
α
1
1
*
3
−z
PSfrag replacements −0.5
5
u2
γ9
α
γ
σ
σ
u1
γ
α
µ
u2
u2
0.5
u2
Sfrag
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
y
u
α
**
−z
µ
−1
3 −0.5
5
9 −1
*
−z
−0.5
0
u1
0.5
1
α
−1
(c) γ = 0.15
−0.5
0
u1
(d) γ = 0.01
Figure 4.11: Isoclines for α = 0.7 and different values of γ.
(a) h1 (0) = (γ1− , 0, 0, 0),
positive on s ∈ (0, 1).
(b) h2 (0) = (γ3− , 0, 0, 0),
h1 (1) = (γ3+ , 0, 0, 0), and
∂γ
∂s
is continuous and
h2 (1) = (γ1+ , 0, 0, 0).
(c) h1 (·) and h2 (·) represent all the symmetric equilibria of the discrete SH
equation, up to symmetry. More specifically, let (γ, u) be a nontrivial symmetric equilibrium of the discrete SH equation. Then, there exists a unique
s ∈ (0, 1) and a unique i ∈ {1, 2} such that either hi (s) = (γ, u) or
hi (s) = (γ, −u).
The local results of this proposition can be proved analytically, by a simple
127
bifurcation argument. When the local results are obtained, then the Implicit
Function Theorem states that the local result can be extended to an interval
where the appropriate determinant of the Jacobian matrix does not vanish. This
condition is checked numerically.
4.5
Dynamics
In the previous section we have obtained a complete description of the set equilibrium solutions of the discrete Swift-Hohenberg equation for different values
of α and γ. Knowing that solutions of the discrete Swift-Hohenberg equation
converge to one of these equilibrium solutions, we will ask in this section which
particular equilibrium solution will be selected. In [53] and [54] this same question was discussed for the Swift-Hohenberg equation. The phenomena observed
there were in part a motivation for the study presented in this paper.
Throughout this section we shall assume that the initial value u0 is symmetric, and hence the solution u(t) will be symmetric for all time. Thus, we shall
be studying solutions of the system (4.38):
(
u01 = (−6γ 2 + 4γ)u1 + (4γ 2 − 2γ)u2 − (1 − α)u1 − u31 ,
(4.43)
u02 = (8γ 2 − 4γ)u1 + (−6γ 2 + 4γ)u2 − (1 − α)u2 − u32 .
If 0 < α < 21 , then for any given γ > 0, the set of equilibrium solutions is very
limited. It consists of the origin and, for some values of γ, of a pair of nontrivial
solutions.
If 21 < α < 1, then, depending on the value of γ, there are many more nontrivial
equilibrium solutions. Therefore, here we shall focus on this case. First, let us
exhibit the critical values found in the previous section for α = 0.7:
γ3− ≈ 0.04784,
γ∗+
≈ 0.51651,
γ1− ≈ 0.27884,
γ3+
≈ 0.53795,
γ∗− ≈ 0.47272,
γ1+ ≈ 3.13537.
(4.44)
As in [53] and [54], we begin with a series of numerical experiments, and
follow orbits for different values of γ. We shall firstdo this for
π
π 1 u0 = sin
, sin
= √ ,1
and
α = 0.7.
(4.45)
4
2
2
In [53] and [54] the large time behaviour of solutions was studied for increasing
values of L. Thus, to emulate this analysis we consider a decreasing sequence of
128
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
1
2
x
3
4
1
1
2
x
3
−0.5
1
3
5
0
−0.5
1
2
x
3
(d) γ = 0.47
4
α
1
2
x
3
4
1
0.5
0
−0.5
9
−1
0
−1
0
4
0.5
u
3
5
0
9
α
1
1
0.5
u
3
5
−0.5
−1
0
u
−1
0
1
u
1
u
u
frag replacements
PSfrag replacements
PSfrag replacements
γ
γ
γ
α
α
α
values
of γ = 1/L2 . In Figureγ4.12 we show solution profiles
γ
γ corresponding to six
values of γ: γ = 4, 3, 0.48, 0.47, 0.05 and 0.01. Given the initial condition u0 ,
σ
σ
σ
we will denote the orbit by φt (u0 , γ) and the final equilibrium by φ∞ (u0 , γ). We
σ
σ
σ
plot
u1 the initial profile u0 byua1 solid line, a few intermediate
u1 profiles by dashed
lines
γ and the final profile φ∞γ(u0 , γ) by a thick line.
γ
α
α
α
µ
frag replacements
PSfrag replacementsµ
PSfrag replacementsµ
1
γ
γ1
γ1
3
3
α
α
α3
5
γ
γ5
γ5
9
9
σ
σ
σ9
ασ
α
σ
σα
u1
u1
u1
(a) γ = 4
(b) γ = 3
(c) γ = 0.48
γ
γ
γ
α
α
α
µ
µ
µ
9
−1
0
1
2
x
3
(e) γ = 0.05
4
α
−1
0
1
2
x
3
4
(f) γ = 0.01
Figure 4.12: Dynamics for α = 0.7 and different values of γ.
It is evident that the final profile φ∞ (u0 , γ) depends critically on the value
of γ. In order to follow the change in φ∞ (u0 , γ)
more accurately as γ changes,
we show in Figure 4.13 the action J φ∞ (u0 , γ) as γ varies, again for the choice
of u0 and α given in (4.45).
We see that for γ = 4 the orbit converges to the origin. Indeed, since
γ1+ = 3.135... for α = 0.7, we know that for γ = 4 the origin is the only
equilibrium state. We see that when γ = 3, and hence has dropped below γ1+ ,
the final state is no longer the trivial solution, but the unique solution (v1 , v2 ),
which lies in the first quadrant, and on the branch of solutions which bifurcates
subcritically from the trivial solution at γ1+ . This equilibrium was denoted in
the previous section by u. As γ is reduced further, we see that around γ = 0.47,
the graph of J φ∞ (u0 , γ) exhibits a discontinuity, and that φ∞ (u0 , γ) jumps
129
Js
1
3
−0.1
γ
α
γ
σ
σ
u1
γ
α
µ
−0.15
1
0
−0.05
−0.2
3
−0.25
5
9
α
−0.3
−0.35
−0.4
0
0
−0.05
−0.1
Js
γ
α
γ
σ
σ
u1
γ
α
µ
0.5
1
1.5
2
2.5
3
3.5
`
´
(a) Js φ∞ (u0 , γ) vs. γ
−0.15
5
9
−0.2
α
−0.25
0.4
0.45
0.5
0.55
0.6
`
´
(b) Js φ∞ (u0 , γ) vs. γ,
amplified
Figure 4.13: Js φ∞ (u0 , γ) vs. γ for the choice (4.45)
from a profile in the first quadrant at γ = 0.48 to one which lies in the fourth
quadrant at γ = 0.47. This equilibrium was denoted in the previous section by
−z∗ . As γ is reduced further, φ∞ (u0 , γ) remains in the fourth quadrant and
becomes the trivial solution again at γ = γ3− . In Figure 4.14 we can clearly
view the jump of the final profile φ∞ (u0 , γ). There we draw in the phase plane
the isoclines (in solid line) and the orbits (in thick line) followed by solution of
equation (4.43) for the choice for α and u0 given by (4.45) and two different
values of the parameter γ : 0.47 and 0.48.
We will introduce some technical definitions that are motivated by our numerical simulations in order to describe the dynamics and, more specifically, the
variation of φ∞ (u0 , γ) with respect to γ and u0 . In the next section, we will
make phase plane analysis. It will shed some light and help us to understand
the previous simulations.
As we have seen, the final profile φ∞ (u0 , γ) varies with γ. Specifically, let
u0 be fixed and consider the map
γ → φ∞ (u0 , γ),
for γ > 0.
(4.46)
Then this map is piecewise continuous and for every u0 ∈ 2 , u0 6= (0, 0), the set
of discontinuities is nonempty. This allows us to define the set of discontinuities
of the map φ∞ (u0 , ·) and order this set. Therefore, we define
def
S(u0 ) = {γ̂k : φ∞ (u0 , ·) is discontinuous at γ = γ̂k } .
130
(4.47)
1
3
−0.5
1
3
0.5
5
9
α
1
0.5
u2
0
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
u2
frag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
0
−0.5
5
−1
−1
9
−0.5
0
u1
0.5
1
α
(a) γ = 0.48.
−1
−1
−0.5
0
u1
0.5
1
(b) γ = 0.47.
Figure 4.14: Graphs of the isoclines and orbits for symmetric discrete SH equation and conditions given by (4.45)
We define the set S(u0 ) in such a way that the discontinuity values γ̂k are
ordered increasingly. We know that S(u0 ) is nonempty and it could consist of
one value.
The set S(u0 ) varies with u0 . In the next plot we show this; we have
chosen α = 0.7 but as initial data we have now chosen u0 = (1, 51 ). We show
that in this case the discontinuity in the final equilibrium φ∞ (u0 , ·) occurs for
some value γ̂ ∈ (0.48, 0.485), but that φ∞ (u0 , γ) is continuous on the interval
(0.47, 0.48), while in the previous case (4.45) a discontinuity occurred on the
interval (0.47, 0.48). We also note that the jump occurs at some value slightly
higher or equal to γ∗− and lower than 21 .
Let us finish by a lemma that can help us to describe the dynamics and
determine the final profile in some sets. We define the quadrants
def Q1 =
(u1 , u2 ) ∈ 2 : u1 > 0 and u2 > 0 ,
def Q2 =
(u1 , u2 ) ∈ 2 : u1 < 0 and u2 > 0 ,
def Q3 =
(u1 , u2 ) ∈ 2 : u1 < 0 and u2 < 0 ,
def Q4 =
(u1 , u2 ) ∈ 2 : u1 > 0 and u2 < 0 .
Lemma 4.4
1. For γ > 21 , the quadrants Q1 and Q3 are positively invariant.
131
γ
α
γ
σ
σ
u1
γ
α
µ
1
0.5
0
3
5
−0.5
9
α
0.5
0
3
5
−0.5
9
−1
−1
−0.5
0
u1
0.5
(a) γ = 0.485.
1
α
1
0.5
1
u2
3
5
1
1
u2
1
γ
α
γ
σ
σ
u1
γ
α
µ
u2
γ
α
γ
σ
σ
u1
γ
α
µ
0
−0.5
9
−1
−1
−0.5
0
u1
(b) γ = 0.48.
0.5
1
α
−1
−1
−0.5
0
u1
0.5
1
(c) γ = 0.47.
Figure 4.15: Graphs of the isoclines and orbits for α = 0.7 and u0 = 1, 51
2. For γ < 21 , the quadrants Q2 and Q4 are positively invariant.
This lemma is easily proved by inspection of the vector field on the boundary
of the quadrants. Note that in each quadrant we have at most one stable
equilibrium. Therefore, this lemma can determine the final profile φ∞ (u0 , γ)
for different values of γ and u0 and the discussion is considerably reduced.
In the next section we will try to give the global picture of the dynamics
and reformulate the question about the discontinuities of the map in (4.46).
Theorem 4.7 will be an important ingredient in the discussion.
4.6
Phase plane analysis and large time behavior
Let us describe what we are going to do. First, we recall Theorem 4.7 and the
critical values found there. We study the different intervals and give a global
picture of the dynamics via a phase-plane argument. Then, we make a few
observations and we formulate the claim of Proposition 4.3. This allows us to
give a different characerization of the set S(u0 ) of discontinuities of the map
(4.46).
We want to emphasize that the numerical simulations of last section have
been the main motivation for this one. The section has been devoted to describe
some relevant phenomena, in particular, the dependence of φ∞ (u0 , γ) with respect to u0 and γ and the observed discontinuities. Our intention is mainly
descriptive and our approach uses both numerical and analytical arguments.
132
σ
u1
γ
α
µ
1
3
Therefore,
we do not pretend to prove rigorously all the statements contained
5
here in
the
form of “hard” theorems (although we claim it can be done) but
9
understand
and
describe what is observed and give an outline of the results that
α
stand behind the simulations.
For convenience, let us rewrite the sequece of critical values found in Theorem 4.7: The numerical values were given in (4.44).
(a)
0
(b)
γ3−
(c)
γ1−
(d)
γ∗−
(d)
1/2
(c)
γ∗+
(b)
γ3+
(a)
γ1+
Figure 4.16: Critical values of γ.
(a) γ ∈ (0, γ3− ) ∪ (γ1+ , ∞).
By Theorem 4.7, the origin is globally stable. We plot the stable equilibrium
by a circle and the isoclines by dashed lines.
(b) γ ∈ (γ3− , γ1− ) ∪ (γ3+ , γ1+ ).
By Theorem 4.7, the origin is a saddle and we have two stable equilibria
P1 and P2 . We consider the stable manifold of the origin Ws (0)(γ). As we
know, for fixed γ, Ws (0) is a separatrix, so that the complementary set
2
− Ws (0) = U1 ∪ U2 ,
where U1 and U2 are two disjoint open sets, each of them containing one
different stable equilibrium. Each of them is the domain of attraction of the
different equilibria. This is illustrated in Figure 4.17(b). We plot the stable
equilibria by circles, the saddle by a triangle, the isoclines by dashed lines
and the separatrix by a thick line.
(c) γ ∈ (γ1− , γ∗− ) ∪ (γ∗+ , γ3+ ).
By Theorem 4.7, the origin is an unstable node and we have two stable
equilibria P1 , P2 and two saddles P3 , P4 . Consider the set Ws (P3 )∪Ws (P4 ).
Again, the complementary set satisfies
2
− Ws (P3 ) ∪ Ws (P4 ) = U1 ∪ U2 ∪ {(0, 0)} ,
where U1 and U2 are two disjoint sets, each of them being the domain of
attraction of one stable equilibrium. This is illustrated in Figure 4.17(c).
133
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
u2
u2
0.5
0
1
3
replacements −0.5
5
−1
−1
PSfrag
−0.5
0
u1
1
0.5
0
1
3
replacements −0.5
5
0.5
γ9
α
γ
σ
σ
u1
γ
α
µ
1
(a) γ = 0.01
1
0.5
u2
γ9
α
γ
σ
σ
u1
γ
α
µ
u2
PSfrag
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
0
−1
−1
0.5
1
α
(c) γ = 0.47
0.5
1
0
3 −0.5
5
9 −1
0
u1
1
0.5
3 −0.5
5
9 −1
−0.5
0.5
1
1
−1
0
u1
(b) γ = 0.2
1
α
−0.5
−1
−0.5
0
u1
(d) γ = 0.48
Figure 4.17: Separatrixes for α = 0.7. Circles represent stable nodes and triangles represent saddles.
(d) γ ∈ (γ∗− , γ∗+ ).
By Theorem 4.7, the origin is an unstable node and we have four stable
equilibria P1 , P2 , P3 , P4 and four saddles P5 , P6 , P7 , P8 . As before, we
consider the set ∪8i=5 Ws (Pi ). The complemantary set satisfies
2
−
8
[
Ws (Pi ) =
4
[
i=1
i=5
Ui ∪ {(0, 0)} .
where the sets Ui are disjoint, each of them being the domain of attraction
of Pi for each 1 6 i 6 4. This is illustrated in Figure 4.17(d)
134
For γ > 0, we define the map
[
def
γ → Λ(γ) =
{Ws (Pi ) : Pi is a saddle } .
(4.48)
i
In Figure 4.17 we represent with a thick line the set Λ(γ) for different values of
γ and fixed α = 0.7.
We will now reformulate the question about the discontinuity of the map
(4.46) and describe the set S(u0 ), defined in (4.47), in terms of the set Λ(γ).
Let u0 ∈ 2 , u0 6= (0, 0) be fixed. For any γ > 0, if u0 ∈ Ws (Pi ) for
some saddle Pi , then φ∞ (u0 , γ) = Pi . Otherwise, there exists a unique Ui
such that u0 ∈ Ui , the domain of attraction of some stable equilibrium Pi , and
φ∞ (u0 , γ) = Pi .
Let γ0 > 0 be a noncritical value, u0 6= (0, 0) be fixed and suppose that
u0 ∈ Ui for some index i. Then, u0 ∈ Ui for all γ ∈ (γ0 − ε, γ0 + ε) for some
ε > 0. Therefore, the map (4.46) is continuous at γ = γ0 . In other words,
leaving the critical values apart, we can characterize the set S(u0 ) as follows
Proposition 4.3 Let γ be a noncritical value. Then, we have
γ ∈ S(u0 ) if and only if u0 ∈ Λ(γ).
Let γ0 be fixed. Since the domain of attraction Ui of a stable node is open,
the map (4.46) is continuous at γ = γ0 for any u0 ∈ Ui . Hence, a discontinuity
can only occur at points u0 contained in a stable manifold of a saddle. To prove
the other implication, we need to prove that S(u0 ) is a set of isolated values. A
sufficient condition is that for every noncritical γ > 0 and every u ∈ Λ(γ), the
d
vector flow dγ
Λ(γ) is transversal to Λ(γ).
Proposition 4.3 gives us an alternative way to describe S(u0 ), by checking if
u0 ∈ Ws (Pi ) for some saddle Pi . The definition is more handy when u0 is fixed
and γ varies; the alternative is more handy when γ is fixed. In the latter case,
we study the evolution of a finite number of orbits, that is, the stable manifolds
of the saddles, with respect to γ.
Proposition 4.3 deals with noncritical values of γ. Let us consider the critical
ones. We will distinguish a first group of critical values γ3− , γ1− , γ3+ , γ1+ , for
which the spectrum of the linearised operator around the origin contains the
zero eigenvalue, and a second group consisting of γ∗− and γ∗+ . This latter group
corresponds to the critical values for which a saddle-node bifurcation occurs, and
the number of stable equilibria changes when γ crosses them, in contrast to the
135
first group. The reason for this distinction can become apparent by looking at
the series of plots in Figure 4.17. The main difference is found in the evolution
of the separatrix. We will try to describe it now.
Proposition 4.4
1. Proposition 4.3 holds for γ ∈ γ3− , γ1− , γ3+ , γ1+ .
2. Proposition 4.3 does not hold for γ ∈ {γ∗− , γ∗+ }.
Let us start with the first group. We recall that these are the critical values for
which the spectrum of the linearised operator around the origin contains the
zero eigenvalue. In fact, for these critical values, Proposition 4.3 can be shown
to hold.
Let us consider first the case γ = γ3− and γ = γ1+ . The evolution can be
observed by comparing Figures 4.17(a) and 4.17(b). For γ < γ3− and γ > γ1+ ,
the set Λ(γ) is empty. When γ crosses these values, the origin becomes a saddle
and the separatrix arises, but two stable equilibria bifurcate from the origin.
Next, let us consider the case γ = γ3+ and γ = γ1− . The evolution can be observed by comparing Figures 4.17(b) and 4.17(c). When γ crosses these values,
the origin becomes an unstable node and two new saddles bifurcate from the
origin, but the map defined in (4.48) is continuous. Therefore, Proposition 4.3
also holds for these values of γ.
Finally, the values γ∗− and γ∗+ are qualitatively different and that can be
observed by comparing Figures 4.17(c) and 4.17(d). For γ ∈ [γ3− , γ∗− ) ∪ (γ∗+ , γ1+ ]
Λ(γ) consists of two orbits, while for γ ∈ [γ∗− , γ∗+ ], Λ(γ) consists of four orbits.
Let γ ∈ [γ∗− , γ∗+ ] and for every 1 6 i 6 4 define Pi (γ) as the unique stable
node found in the i-th quadrant Qi and let Ui (γ) be its domain of attraction.
Then, we have U1 (γ∗− ) = limγ&γ∗− U1 (γ). Let u0 ∈ U1 (γ∗− ). Then there exists
some ε > 0 such that
φ∞ (u0 , γ) = P4 (γ)
φ∞ (u0 , γ) = P1 (γ)
for every
for every
γ ∈ (γ∗− − ε, γ∗− ) and
γ ∈ [γ∗− , γ∗− + ε).
In particular, γ∗− ∈ S(u0 ) and u0 is not contained in the stable manifold of any
saddle, so Proposition 4.3 does not hold for γ∗− . A similar argument can be used
for γ∗+ .
We can see this last property in Figure 4.18. We have plotted the orbit for
equation (4.43), with
1
3
α = 0.7 and initial data u0 = 1,
, and u0 = 1,
(4.49)
2
4
136
PSfrag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
1
0.5
u2
u2
rag replacements
γ
α
γ
σ
σ
u1
γ
α
µ
0
1
3 −0.5
5
9
α
−1
−1
1
0.5
0
1
3 −0.5
5
9
−0.5
0
u1
0.5
1
α
(a) γ = 0.475.
−1
−1
−0.5
0
u1
0.5
1
(b) γ = 0.47.
Figure 4.18: Graphs of the orbits for equation (4.43) with conditions given by
(4.49).
Both initial data are contained in U1 (γ∗− ) and they are shown by a circle. We
plot the isoclines with dashed lines and the orbits with thick line. For γ = 0.475,
φ∞ (u0 ) = P1 while for γ = 0.47, φ∞ (u0 ) = P4 . Note that this case is different
from the case shown in Figure 4.15, where the initial data was u0 = 1, 51 .
There,
occurs for a noncritical value. The reason is that u0 =
the discontinuity
/ U1 (γ∗− ).
1, 15 ∈
137
138
Appendix A
A.1
Proof of Lemma 2.3
Lemma 2.3 Let u(x) be a nontrivial homoclinic orbit to the origin that solves
the differential equation (2.3) where the nonlinearity f is given by the cubic
polynomial in (2.6). Then
√
2 7
.
(A.1)
q>
3
Proof. We will make use of the interpolation inequality
Z
∞
∞
1
u(x) dx +
u (x) dx 6 λ
4λ
−∞
−∞
0
Z
2
2
Z
∞
u00 (x)2 dx
(A.2)
−∞
which holds for every λ > 0.
Multiply (2.3) by u0 (x) and integrate over x ∈ (−∞, ∞). Then we obtain
the equality
6
Z
∞
−∞
00
2
u (x) dx − 2q
Z
∞
0
−∞
2
u (x) dx −
Z
∞
4
u(x) dx + 2
−∞
Z
∞
u(x)2 dx = 0. (A.3)
−∞
On the other hand, multiplying (2.3) by u(x) and integration over x ∈
(−∞, ∞) gives us
Z
∞
−∞
00
2
u (x) dx − q
Z
∞
−∞
0
2
u (x) dx +
Z
∞
−∞
139
4
u(x) dx −
Z
∞
−∞
u(x)2 dx = 0.
(A.4)
R∞
We can eliminate the term −∞ u(x)4 dx by adding both expressions (A.3)
and (A.4)
Z ∞
Z ∞
Z ∞
1
7
0
2
2
u (x) dx =
u(x) dx +
u00 (x)2 dx.
(A.5)
3q −∞
3q −∞
−∞
A necessary condition about q is that it must be a positive parameter.
3q
. Thus,
Comparison between (A.2) and (A.5) motivates that we take λ = 28
we obtain
Z ∞
Z ∞
Z
Z ∞
7
3q ∞
7
1
2
00
2
2
u(x) dx +
u (x) dx 6
u(x) dx +
u00 (x)2 dx.
3q −∞
3q −∞
28 −∞
3q −∞
(A.6)
or equivalently
Z ∞
3q
1
−
u(x)2 dx > 0.
(A.7)
28 3q
−∞
As u is a nontrivial solution, the bound (A.1) follows.
A.2
Weak convergence
Consider the equations
1 iv
v (y) + v 00 (y) + f (v(y)) = 0,
q2
(A.8)
v 00 (y) + f (v(y)) = 0,
(A.9)
and
where f is the piecewise linear function defined in (2.7). It is known that for
q = qk there exists a positive symmetric homoclinic orbit to the origin that
crosses once the line v = 21 at y = ζ = 3π
4 + o(1). The form of the solution for
|y| < z is
vk (y) = 1 + Ak cos(ak y) + Bk cos(bk y),
(A.10)
where
Ak ∼
1
,
k3
ak ∼ k,
1
Bk ∼ √ ,
2
bk ∼ 1,
Consider V the solution of equation (A.9) with
3π
0
v (0) = 0,
v
=
4
140
as k → ∞.
boundary conditions
1
,
2
(A.11)
in other words,
3π
,
4
3π
y>
.
4

1

 1 + √ cos y
2
V (y) =

1
3π
 e 4 −y
2
|y| 6
(A.12)
It was shown in Lemma 2.10 that vk → V uniformly on C 2 ( ). However, vk000 (y)
does not converge even pointwise.
Call L = 3π
4 and I = (−L, L). The main result is the following:
Theorem A.1 vk000 converges to V 000 in the following weak sense: for every
bounded continuous function h defined on I we have
lim
k→∞
Z
L
−L
vk000 (y)h(y)dy =
Z
L
V 000 (y)h(y)dy.
−L
Proof. To prove this theorem, we note that vk000 and V 000 are given by
vk000 (y)
V 000 (y)
= Ak a3k sin(ak y) + Bk b3k sin(bk y),
1
= √ sin y.
2
(A.13)
(A.14)
Thus, by an application of Lebesgue’s dominated convergence theorem, we have
lim
k→∞
Z
L
−L
Bk b3k sin(bk y)h(y)dy =
Z
L
−L
1
√ sin(y)h(y)dy.
2
It remains to show that
lim
k→∞
Z
L
−L
Ak a3k sin(ak y)h(y)dy = 0,
(A.15)
for every bounded continuous function h on I.
Let α > 0 and consider
fα (x) = sin(αx),
f+,α (x) = max(fα (x), 0),
and call
k = max{n > 0 : n 6
141
f−,α (x) = max(−fα (x), 0),
3α
}.
8
Thus, by the algorithm of division, we have
αL = 2πk + r
with 0 6 r < 2π.
In particular,
k
3
1
3
6 < + .
8
α
8 α
We compute the integral of f+,α over I
Aα
=
=
=
Z
L
f+,α (x)dx =
−L
1
α
Z
αL
f+,1 (y)dy
−αL
Z (2j+2)π !
Z (2j+1)π
k−1
1 X
0
sin(y)dy +
α
(2j+1)π
2jπ
j=−k
Z
Z
1 −2kπ
1 L
+
f+,1 (y)dy +
f+,1 (y)dy
α −L
α 2kπ
k−1
1 X
4k
3
1
1
1
2+O
=
= +O
+O
α
α
α
α
2
α
j=−k
f
(x)
as α → ∞.
Call g+,α (x) = +,α
Aα . Then g+,α (x) is a probability density on I for every
α > 0. We are interested in the behaviour for α → ∞.
142
Let us compute the Fourier transform of g+,α (x).
Z L
F(g+,α )(t) =
eitx g+,α (x)dx
−L
Z (2j+1)π
Z (2j+2)π !
k−1
1 1 X
ity/α
e
sin(y)dy +
0
Aα α
2jπ
(2j+1)π
j=−k
Z
Z
1 1 L
1 1 −2kπ
f+,1 (y)dy +
f+,1 (y)dy
+
Aα α −L
Aα α 2kπ
k−1
X
1 1 α2
1
itπ/α
it2jπ/α
1+e
e
+O
2
2
Aα α α − t
α
=
=
j=−k
2
eit2kπ/α − e−it2kπ/α
1
1 1 α
itπ/α
+
O
1
+
e
A α α α2 − t 2
α
eit2π/α − 1
1
α2 1 + eitπ/α sin(2kπt/α)
1
+O
=
Aα α2 − t2 eitπ/α α sin(πt/α)
α
sin(Lt)
+ o(1) = F(g∞ )(t) + o(1),
Lt
=
=
=
where

 1
g∞ (x) = 2L

0
|x| 6 L
otherwise.
Arguing similarly as before, we proceed to define
Z L
3
1
Bα =
f−,α (x)dx = + O
.
2
α
−L
f
(x)
Call g−,α (x) = −,α
Bα . Again g−,α (x) is a probability density on I for every
α > 0. We compute its Fourier transform
F(g−,α )(t) = F(g∞ )(t) + o(1)
Thus, as α → ∞, both the Fourier transform of g+,α (x) and g−,α (x) tend to
the Fourier transform of g∞ .
By the use of Levy’s Theorem (see, for instance, [8]), we know that for every
bounded continuous function h on I
Z L
Z L
Z L
g∞ (x)h(x)dx.
lim
g+,α (x)h(x)dx = lim
g−,α (x)h(x)dx =
α→∞
−L
α→∞
−L
143
−L
Since
fα = f+,α − f−,α = Aα g+,α − Bα g−,α ,
we conclude that for every bounded continuous function h on I
lim
α→∞
Z
L
fα (x)h(x)dx = 0.
−L
From this last equality, (A.15) follows.
A.3
A.4
Values of qk , ζk and of uk and u00k at the
origin
k
qk
ζ(qk )
u(0)
u00 (0)
1
2
3
4
5
6
7
8
9
10
2.676238584
3.782513691
5.000217193
6.264233336
7.551688580
8.852533501
10.16170325
11.47638465
12.79489648
14.11617480
3.404608234
4.339444241
5.113599136
5.787981648
6.392938791
6.946103324
7.458758840
7.938627940
8.391264245
8.820817639
1.907870236
1.719342998
1.733251153
1.712938707
1.716282398
1.710412301
1.711669783
1.709215602
1.709815095
1.708563923
-.5694105416
-.1321149080
-.1940547773
-.0910032657
-.1142824189
-.0684516828
-.0804604621
-.0546513892
-.0619473193
-.0454182263
Proof of Theorem 4.1
Theorem 4.1 The number of symmetric equilibria of equation (4.5) is given
by:
√
1. If 0 < σ < 2 − 2, then the origin is the unique equilibrium. It is stable.
√
√
symmetric equilib2. If 2 − 2 < σ < 2 + 2, then there exist two nontrivial
√
ria. They bifurcate from the origin at σ = 2 − 2 (supercritical pitchfork
bifurcation) into the first and the third quadrant and they extend to σ =, ∞.
144
√
They are stable nodes for every σ > 2 − 2 and in this regime, the origin
is a saddle.
√
3. If 2 + 2 < σ < σ̂, then there exist four nontrivial equilibrium states.
√ The
two new symmetric equilibria bifurcate from the origin at σ = 2 + 2 (supercritical pitchfork bifurcation) into the second and the fourth quadrant,
√
and they extend to σ = ∞. They are saddles for every σ > 2 + 2 and
the origin becomes an unstable node.
4. If σ > σ̂, then there exist eight nontrivial equilibria. The four new symmetric equilibria emerge at σ = 2 + µ̂ from a saddle-node bifurcation at
a point in the second, respectively the fourth, quadrant. The equilibrium
states with the highest value of |u2 | are stable nodes and the other two are
saddles.
Proof. We are searching for symmetric equilibria of equation (4.5a). That is
equivalent to solving the following system
(
x2 − 2x1 + σf (x1 ) = 0,
(A.16)
2x1 − 2x2 + σf (x2 ) = 0.
Let us make some preliminary observations about system (A.16). The origin
is a solution for every σ. By the odd symmetry of f , if (x1 , x2 ) solves (A.16),
so (−x1 , −x2 ) does, and if we put x1 = 0 in (A.16), we obtain the origin as the
unique solution. This allows us to restrict our search for nontrivial equilibria on
x1 > 0. Moreover, for σ = 0, the origin is the unique solution of system (A.16),
so we can restrict to σ > 0. Therefore, we will search for nontrivial equilibria
of (A.16) in the set
def
U =
σ, x1 , x2 ∈
3
: σ > 0,
x1 > 0 .
In Figure 4.2 we have plotted the isoclines for different values of σ, so that
we can obtain a geometric view of the system.√For small σ > 0, the isoclines
intersect only at the origin. The value σ = 2 − 2 is the bifurcation value since
the slopes of both isoclines equalize. At σ = √
2, both isoclines have negative
slope at the origin and eventually, at σ = 2 + 2, both slopes coincide again.
These cases correspond to equilibria bifurcating from the origin. Finally, at
σ = 2 + µ̂, a saddle-node bifurcation occurs.
Let us give an overview of the strategy we will follow. We will convert
system (A.16) into one single polynomial equation for x1 of degree 8. Then we
145
will transform this equation into a simpler equation (A.19). This is a polynomial
of degree 4 for a new variable p. The transformation of both problems is carried
out in (A.18). Therefore, searching for solutions of equation (A.16) in the set
U is equivalent to searching for solutions of equation (A.19) in the following set
def
V =
(p, µ) ∈
2
: µ > −2,
p+µ>0 .
This problem can be solve analytically: we can find an open set O
such
⊂
that the solutions of (A.19) in V correspond to a graph p, µ(p) defined on
p ∈ O. Finally, we draw the corresponding conclusions in terms of the original
variables x1 , x2 and parameter σ.
We start our procedure by solving the first equation of system (A.16) for x 2
and substitute in the second one. This gives us one single polynomial equation
for x1 .After division by x1 we obtain
σ 4 x81 −3σ 3 (σ−2)x61 +3σ 2 (σ−2)2 x41 −σ(σ−2) 1 + (σ − 2)2 x21 +(σ−2)2 −2 = 0.
(A.17)
To simplify the analysis, we make the following transformation,
µ = σ − 2,
p = σx21 + 2 − σ = (2 + µ)x21 − µ.
(A.18)
Then equation (A.17) is equivalent to
def
g(µ, p) = p4 + µp3 − µp − 2 = 0.
(A.19)
Since we look for solutions in the setU , this implies that we are searching for
pairs (p, µ) such that
µ > −2, p + µ > 0
(A.20)
Note that σ = 2 + µ > 0. Therefore, we search for solutions in V .
We can explicitly solve equation (A.19) for µ. This yields
µ(p) =
2 − p4
.
p3 − p
(A.21)
We search for pairs p, µ(p) ∈ V satisfying equation (A.21) and inequalities
√
def
(A.20).
√ Easy analysis shows that this holds iff p ∈ O = (−∞, − 2) ∪ (−1, 0) ∪
(1, 2).
In Figure A.1 we plot the graph of the function µ(p) in the set p ∈ O. The
dashed plot represents the boundaries of the inequalities (A.20), that is, the
146
frag replacements
γ
α
γ
σ
σ
u1
γ
α
1
3
5
9
α
10
8
6
µ
4
2
–6
–5
–4
p
–3
–2
–1
0
1
–2
Figure A.1: Plot of µ(p) in the admissible set O.
graphs of µ = −p and µ = −2. Recall that by (A.20), only the part of the plot
above them is admissible.
√
√
The graph is as follows: for p ∈ (−∞, − 2), µ(p) decreases from ∞ to 2.
For p ∈ (−1, 0), µ(p) decreases
from ∞ to the local minimum
√ µ̂ and it increases
√
back to ∞. For p ∈ (1, 2), µ(p) decreases from ∞ to − 2.
Let us draw the conclusions in terms of the √
parameter µ. For µ close to −2,
there exist no nontrivial equilibrium. At µ = − 2, the first branch of equilibria
√
bifurcates
from the origin and they are well defined for every µ > − 2. At
√
µ = 2, a second
√ branch bifurcates from the origin and they are well defined
for every µ > 2. At µ = µ̂, two branches emerge from a nontrivial equilibrium
(saddle-node bifurcation) and they are defined for every µ > µ̂.
Note that the value µ̂ can be found as the local minimum for µ(p) in the
interval p ∈ (−1, 0). It can be alternatively found by solving the system:
(
g(µ, p) = 0,
(A.22)
∂g
∂p (µ, p) = 0.
We can give an analytic expression of the bifurcation branches σ, x1 (σ)
with parametric coordinates:
1. First branch:
σ(p) = 2 + µ(p),
x1 (p) = ±
s
p + µ(p)
2 + µ(p)
for 1 < p <
√
2.
2. Second branch:
σ(p) = 2 + µ(p),
x1 (p) = ±
147
s
p + µ(p)
2 + µ(p)
√
for p < − 2.
3. Third branch:
σ(p) = 2 + µ(p),
x1 (p) = ±
s
148
p + µ(p)
2 + µ(p)
for
− 1 < p < 0.
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Samenvatting
Wetenschappers gebruiken differentiaalvergelijkingen om fenomenen uit de natuur
te modelleren. Klassieke partiële differentiaalvergelijkingen zijn vaak van tweede
graad. Tegenwoordig is er groiende belangstelling voor andere soorten nietlineare vergelijkingen, zoals differentiaalvergelijkingen van vierde orde of differentiaalvergelijkingen met vertraging. In het proefschrift bestuderen wij sommige
van die vergelijkingen.
In Hoofdstuk 2 beginnen wij met de Swift-Hohenberg vergelijking, die wordt
gebruikt als een eenvoudig model voor patroonformatie. Het is een ingewikkelde
vergelijking. Wij hebben de niet-lineariteit aangepast. Wij bewijzen dat deze
nieuwe vergelijking oneindig veel puls vormige oplossingen bezit, met monotone
‘staarten’, en dat deze ingebed zijn in families pulsen met periodieke staarten.
In Hoofdstuk 3 bestuderen wij een vergelijking over , de gehele getallen en
zoeken we lopende golven. Daarvoor moeten we een vergelijking met vertraging
oplossen. We vinden monotone golven, onbegensde golven en golven die één
monotone staart en één periodieke staart hebben.
In Hoofdstuk 4 bestuderen we een discreet systeem, voorgesteld als vereenvoudigd model van de klassieke Swift-Hohenberg vergelijking, om de gecompliceerde dynamica van deze vergelijking te begrijpen, en hoe die afhangt van
karakteristieke parameters. Wij analyseren de evenwichtspunten, bifurcaties en
stabiliteit. Uiteindelijk maken we enige numerieke simulaties om de belangrijkste fenomenen te illustreren.
155
156
Afterword
This book has been the result of more than four years of joint work. Besides the
coauthor of the articles, the other members of the research group ‘Differential
Equations’ at Leiden University have created a nice atmosphere of fruitful discussions. Common work with the Lorentz Center of Leiden and the University
of Delft, will be the roots of collaborative work. This work has been supported
financially by the Mathematical Institute, Snellius, and the Thomas Stieltjes
Institute for Mathematics.
Besides the academic work, there are a number of people that have made
this possible by sharing good moments with me. I will start mentioning my
family, my mother Begoña, my sister Mari Carmen, Antonio and Patxi, and our
dog Zeru. The book is dedicated to them. Distance has not avoided keeping
close contact and love. Also my friends in Bilbao have kept my motivation
sufficiently high when it was difficult to do so: Juanma, David, Ibán, Gonzalo,
Alex, Alberto Alcalde, Gaizka, Javi, Alberto Varona, Angel, Gabriel Rodrigo,
Carlos, Gabriel de la Rica, José Manuel, my friends from the church, Rosi,
Saúl, Olga, or my faculty colleagues, Aitor, Pedro and Jesús. I will include
my friends from Groningen: Alex, Montse, Elena, Marga, Marcos, Dani, David,
Tino, Emiliano and Daniele.
From the first week I started my PhD, I received a warm welcome from the
‘clan of Amsterdam’, namely, Alex and Andrea, Juan, Manuel, Jorge, Jorge
Miguel, Jaime and his big family, Marc and Carolina. I spent more than a year
living together with the extraordinary couple Paul and Jokke, and my flatmates
there. I got enthusiastic singing in the Leiden English Choir and sharing drinks
in the ‘AiO borrel’.
At the university, I have shared ‘gezellige’ lunches and dinners with other
PhD students: Miguel, the patient Luca, Sofı́a, Guillermo, Leila, Bart, Federica,
Bernadetta, Szabolcs, Derk, Josep. I have had many meetings with the secretary
Gonnie, who has helped me so much with all the bureaucracy.
157
In St. Agneskerk I have met fantastic people: Juan, Francisco and Henry,
my friends Vilma, Marco, Nhayibe, Serafina, together with all the members of
the third community of St. Agneskerk. We have shared many special moments
and experiences. Finally, I want to keep a special space for my flatmates, my
‘brother’ Miguel, his wife Marilia and Juliano. It is impossible to summarize in
such a short space how important their support and friendship have been.
I will keep them in my mind wherever I go.
158
Curriculum vitæ
José Antonio Rodrı́guez Conde was born on January 23, 1976, in Baracaldo,
at the province of Vizcaya, in Spain. He studied at Miguel de Unamuno high
school, in Bilbao, and he finished it in 1994. He got his Master’s degree of
Mathematics at the university of the Basque Country, in Spain, in 1999. He
studied one of the 5 years at the University of Groningen (Netherlands), with
an Erasmus scholarship from the EU. He started his PhD in Mathematics at
the University of Leiden, under the supervision of prof. dr. ir. L.A. Peletier.
He joined the ‘Analysis and Dynamical Systems’ research group and this book
is the result of this effort.
As a member of the research group, he has attended the discussions of the
group and given presentations of his results. He has attended the course ‘Dynamical Systems and Nonequilibrium Pattern formation’, given at the Lorentz
Center, in Leiden. He has taught exercise classes in the following courses: Differential Equations, Partial Differential Equations, Numerical Analysis.
Apart from work, he has been involved in several social movements. He
has been a monitor for a Christian group of teenagers called ‘Sustraiak’, which
means ‘roots’. He has helped the nuns ‘Damas apostólicas’ to serve food to
homeless people. He is a member of the Association for Peace in the Basque
Country. He likes music, specially classical music. He has studied piano for 6
years and he has sung in two different choirs in Bilbao and the English Choir
of Leiden. He likes reading, specially novels and books about history and also
writing.
In The Netherlands, he has followed Dutch courses. He is involved in several
activities around the catholic church called St. Agneskerk in The Hague, or the
church of St. Agnes. There he teaches English lessons to Latin American inmigrants. Currently he is a member of a Christian neocathecumenal community.
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