Z. Angew. Math. Phys. 66 (2015), 607–629
c 2014 Springer Basel
0044-2275/15/030607-23
published online May 3, 2014
DOI 10.1007/s00033-014-0422-9
Zeitschrift für angewandte
Mathematik und Physik ZAMP
Wave speeds for the FKPP equation with enhancements of the reaction function
Freddy Dumortier and Tasso J. Kaper
Abstract. In classes of N -particle systems and lattice models, the speed of front propagation is approximated by that of the
corresponding continuum model, and for many such systems, the rate of convergence to the continuum speed is known to
be slow as N → ∞. This slow convergence has been captured by including a cutoff function on the reaction terms in the
continuum models. For example, the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with a cutoff has fronts
2
π
that travel at the speed c ∼ cFKPP − (ln(N
, which agrees well with data from numerical simulations of the corresponding
))2
N -particle systems, where cFKPP is the linear spreading speed. In Panja and van Saarloos (Phys Rev E 66:015206, 2002),
an example is presented in which a small enhancement of the reaction function causes the propagation speeds of fronts to
be larger than cFKPP . Such front speeds are also observed in stochastic lattice models where the growth rates in the regime
of few particles are modified. In this article, we analyze the dynamics of traveling fronts in the FKPP equation with the
constant enhancement function employed by Panja and van Saarloos. We present formulas for the wave speeds, develop
the criteria on the parameters for which the front speeds are larger than the linear spreading speed even in the limit in
which the size of the cutoff domain vanishes, study the rate of approach as N → ∞, and identify the mechanisms in phase
space by which the constant enhancement of the reaction function makes possible the larger than linear wave speeds. In
addition, we extend these results to the FKPP equation with two other enhancement functions, which are also of interest for
continuum level modeling of lattice models and many-particle systems in the regimes of small numbers of particles, namely
a linear enhancement function and an enhancement that is uniform above the linearized reaction function. We also derive
explicit formulas for the parameters in these problems. The mathematical techniques used herein are geometric singular
perturbation theory, geometric desingularization, invariant manifold theory, and normal form theory, all from dynamical
systems.
Mathematics Subject Classification (1991). 35K57 · 34E15 · 34E05.
Keywords. Reaction–diffusion equations · Enhancement: Cutoff · Traveling waves · Blow-up technique.
1. Introduction
The Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation
∂2u
∂u
=
+ u(1 − um ),
(1)
∂t
∂x2
with m = 1, 2 [26,31] arises ubiquitously as a model in biology, combustion, ecology, interacting particle
systems, optics, phase transitions, physiology, plasma physics, and other fields, see e.g., [2–4,11,28]. In
these problems, general data often evolve toward traveling fronts, which are solutions that are stationary
in a frame moving at a constant speed c > 0 and that connect the spatially homogeneous states u = 0
and u = 1 ahead of and behind the front, respectively. For example, the FKPP equation may be derived
in the large-scale limit of interacting particle systems or in the mean-field limit of microscopic or lattice
models, and in these systems, the fronts travel with speeds approximated by the classical FKPP speed
cFKPP = 2, see e.g., [8–10,14,15,29,30]. Hence, considerable research has been devoted to finding fronts
and traveling waves in reaction–diffusion models, as well as to determining their propagation speeds.
For broad classes of interacting particle systems, the characteristic propagation speed is known to
converge only slowly to the value predicted by the continuum limit as the number of particles, or average
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F. Dumortier and T. J. Kaper
ZAMP
occupancy number, N → ∞. Even for large values of N such as N = 106 , the propagation speeds
are substantially smaller than expected, as has been observed in systematic numerical simulations, see
[9,10,14,29,30,34]. This slow convergence as N grows has been captured successfully in the context of
continuum models by using cutoff functions for the reaction terms. Indeed, the pioneering analysis in [12]
introduced a Heaviside cutoff function in the FKPP equation to set the reaction function to zero when and
wherever the concentration u decreases below a threshold. The motivation for introducing the Heaviside
cutoff was that few or no particles are present at points in the domain at which the concentration is
less than ε = N1 , and, hence, at these points, few or no reactions take place. For the FKPP (1) with f
set to zero on the interval 0 ≤ u ≤ ε = 1/N , they derived asymptotics showing logarithmic (i.e., slow)
convergence to the continuum limit:
c∼2−
π2
(ln ε)2
as ε → 0.
(2)
Moreover, it was shown in [12] that this logarithmic convergence agrees well with the data from sequences
of numerical simulations with successively larger values of N .
Since publication of [12], this slow convergence in the large-N limit has been analyzed for several
classes of problems with pulled and fluctuating fronts, see, for example, Section 7.1 of [42]. In addition,
mathematical analysis of the corresponding continuum models with cutoff functions has been presented
in [5,6,19,21,22,35], where it was shown that the first-order correction in (2) is universal for pulled fronts
within a large class of cutoff functions and where the impact of cutoffs was also studied on bistable fronts
and pushed fronts in other prototypical reaction–diffusion equations, such as the Nagumo equation.
More recently, a particular case of the FKPP equation with cutoff has been studied in which the
propagation speed is substantially larger than cFKPP , and in which the speed remains larger than cFKPP
even in the limit as ε → 0. In [36], Panja and van Saarloos introduced an enhancement function to the
reaction function f on a narrow interval of concentration values adjacent to the cutoff threshold, and they
showed that —with this enhancement—the propagation speed c(ε, α) = c0 (α)+c̃(ε, α) has c0 (α) > cFKPP ,
with c̃ = o(ε), where α is a parameter that controls the width of the enhancement interval, see (4) below.
These problems are of interest, because these larger wave speeds have been observed in stochastic lattice
models which have modified growth rates in the regimes of low particle concentrations, see [36].
In this article, we study the FKPP equation (1) with the enhancement of [36]:
∂2u
∂u
=
+ f (u, ε, α),
∂t
∂x2
where the reaction function f is defined in the following piecewise manner:
⎧
if 0 ≤ u ≤ ε,
⎪
⎨0
if ε < u ≤ αε,
f (u, ε, α) = εα
⎪
⎩
m
u(1 − u ) if αε < u,
(3)
(4)
with α ≥ 1 and m = 1, 2, . . .. We label the three regions as Regions 1–3, respectively. Region 1, u ∈ [0, ε],
corresponds to the by-now standard cutoff region, and we have taken the Heaviside cutoff for convenience
in the calculations. Several of the key quantities that enter into the formulas for the wave speeds and into
the determination of the critical width of the enhancement interval depend on the choice of the cutoff
function that is used in Region 1. Extensions to other cutoffs of the type used in [12,21] are possible
following the same type of explicit calculation presented here. Region 2, u ∈ (ε, αε], corresponds to the
new interval of enhancement introduced in [36], and Region 3, αε < u ≤ 1, corresponds to the bulk of
the domain in which the FKPP reaction function f is unmodified. We note that there are discontinuities
in f at u = ε and u = αε; see also the proposition in Sect. 3.1.
We establish the following theorem, which rigorously justifies some of the calculations for (3)–(4)
presented in [36] and which presents additional results about the convergence of the speeds to the limiting
speeds. The analysis also identifies the geometry responsible for the existence of the critical value α∗ ; see
Vol. 66 (2015)
Wave speeds for the FKPP equation
609
also Remark 4. Moreover, the proof of the theorem also identifies the mechanisms in the phase space that
are responsible for making the larger speeds possible. When Region 2 is sufficiently wide, the enhancement
causes the fronts to become pushed.
Theorem 1. For each α ≥ 1 and for each m = 1, 2, . . ., there exists an ε0 = ε0 (α, m) > 0 such that
for ε ∈ (0, ε0 ), the reaction–diffusion equation (3) with enhancement (4) has a unique traveling wave
4e2
solution. There also exists a critical value of α given by α∗ = 1+e
2 such that the speeds c(ε, α) of the
traveling waves are as follows: For α < α∗ ,
π2
(5)
+ O (ln ε)−3 .
c(ε, α) = 2 −
2
(ln ε)
For α = α∗ ,
π2
(6)
+ O (ln ε)−3 .
2
4(ln ε)
For α > α∗ , there exists a monotonically increasing, analytic diffeomorphism c0 : (α∗ , ∞) → (2, ∞) such
that the propagation speed is given by
c(ε, α) = 2 −
c(ε, α) = c0 (α) + c̃(ε, α).
(7)
The function c0 (α) is the unique solution of the equation α = α0 (c), where
α0 (c) =
1− 1−
c2
2
+
√
c
2
c2
,
c √2
c2 − 4 e− 2 (c+ c −4)
(8)
0
and dα
dc (c) > 0 for c > 2, so that c = c0 (α) is a simple root of α = α0 (c). Also, there exists a monotone,
analytic diffeomorphism β0 : (2, ∞) → (0, ∞) with
1
β0 (c) = c2 − 4 + c c2 − 4 ,
(9)
2
such that for any c ∈ (2, ∞) and for any m = 1, 2, . . ., the following properties hold for the correction
function c̃(ε, α):
⎧
β0 (c)
)
if c < √m+2
,
⎪
⎪ O(ε
m+1
⎨
m+2
m
c̃(ε, α0 (c)) = O(ε ln(1/ε)) if c = √m+1 ,
(10)
⎪
⎪
m+2
⎩ O(εm )
if c > √
,
m+1
where the symbol O is used in the strict sense.
For α < α∗ , the result of this theorem follows from the calculations performed in [12,21]. The enhancement is insufficiently large to cause any change in the propagation speed. Moreover, we observe that the
first-order correction term is universal, as was shown in [21] in the absence of enhancement functions,
though the value of α∗ depends on the choice of the cutoff function. Then, for α = α∗ , the enhancement
is only large enough to alter the first-order asymptotic correction.
In the most interesting case, α > α∗ , the enhancement is sufficiently large to increase the leading-order
wave speeds, c0 (α), above the linear spreading speed cFKPP = 2. This is the primary impact and interest
of this example. In the course of finding the formula for c0 (α) and demonstrating that c0 (α) > cFKPP for
α > α∗ , we also show that the function α0 (c) extends continuously down to c = 2, with α0 (2) = α∗ and
that 1 ≤ α0 (c) < c2 for all c ∈ [2, ∞). Finally, we observe that the resonances in the normal form of the
governing equations give rise to the fractional powers and logarithmic dependence of c̃(ε, α) on ε.
Remark 1. The value of α∗ is the same critical value as obtained in [36], translated to their parameter
r = 1/α. For α > α∗ , the formula for c0 (α) differs in one term from that in equation (7) in [36]. We note
that there may be a small error in the calculation in [36]. Also, the expressions for c̃(ε, α) differ, as there
are subtleties about the function c̃(ε, α) which the geometric approach here captures naturally.
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F. Dumortier and T. J. Kaper
ZAMP
Traveling waves of (3) with enhancement (4) satisfy the system
U = V,
(11)
V = −cV − f (U, ε, α),
where U (ξ) = u(x, t), ξ = x − ct, and the prime denotes differentiation with respect to ξ. They correspond
to heteroclinic orbits in (11) which connect the homogeneous states U = 1 and U = 0 behind and ahead
of the front,
lim (U, V )(ξ) = (1, 0) ≡ Q−
ξ→−∞
and
lim (U, V )(ξ) = (0, 0) ≡ Q+ .
ξ→∞
In the limit ε → 0, this equation is degenerate at Q+ due to the cutoff in Region 1 and the enhancement
in the reaction function f in Region 2. To remove the degeneracy at Q+ in the traveling wave ODE
system (11), we apply the method of geometric desingularization, also known as the blow-up method. It
expands the degenerate fixed point into a topological sphere, removing the degeneracy and making all
of the fixed points on this sphere hyperbolic or semi-hyperbolic. The orbits of the desingularized vector
field will be tracked in Regions 1–3 on and near this sphere in the appropriate coordinate charts. We
remark that this same coordinate change was also used in [19,21,22,38,39] in the analysis of the impact
of cutoffs on pulled fronts, bistable fronts, and pushed fronts. Hence, this study may also be viewed as a
natural extension of those earlier works.
For the FKPP equation (3) with the constant enhancement (4), each traveling front will then be
obtained as a heteroclinic orbit of the desingularized vector field. In particular, for each (α, m) with
α ≥ 1 and m = 1, 2, . . ., we construct a locally unique singular heteroclinic solution that connects Q−
and Q+ . This singular heteroclinic corresponds to the ε = 0 limit of the desired heteroclinic orbit of
the full system, and it is independent of m. This limiting analysis directly yields the value of α∗ , and
it establishes the properties of the singular solutions, as well as formula (8) for α0 (c). Then, for each
(α, m) with α ≥ 1 and m = 1, 2, . . ., we establish the persistence of the singular heteroclinic for ε > 0
and sufficiently small. The persistence proof also establishes the uniqueness of the traveling wave for each
(α, m) and the asymptotic formulas for c̃(ε, α) for each (α, m), completing the proof of Theorem 1 for
the constant enhancement function in Region 2 introduced in [36].
After completing the proof of Theorem 1, we extend the above results to the FKPP equation (3) with
two different types of enhancement functions in Region 2. We consider a linear enhancement function, in
which the reaction function f is given in Region 2 by
fLE (u, ε, α, γ) = (1 + γ)u
with
γ > 1,
(12)
and a uniform enhancement function, in which the reaction function f is given in Region 2 by
fUE (u, ε, α, δ) = u + εδ
with
δ > 1,
(13)
which represents a uniform enhancement above the linearized reaction function. These different enhancements are useful for other types of modifications of the growth rates used in the few-particle regimes of
stochastic lattice models, microscopic models, and interacting particle models.
For the FKPP equation (3) with the linear enhancement function fLE (12), we prove.
Theorem 2. For each (α, γ, m) with α ≥ 1, γ > 1, and m = 1, 2, . . ., there exists an ε0 = ε0 (α, γ, m) > 0
such that for ε ∈ (0, ε0 ), the reaction–diffusion equation (3) with enhancement (12) has a unique traveling
wave solution. There exists a critical value of α given by
1
1
1+γ
exp √ arctan √
α∗γ = α∗ (γ) =
(14)
γ
γ
γ
such√that for each γ > 1 there exists an analytic, monotonically increasing diffeomorphism cγ0 : (α∗γ , ∞) →
(2, 2 γ + 1) such that the speeds c(ε, α, γ) of the traveling waves are as follows: For α < α∗γ , c(ε, α, γ) is
given by (5). For α = α∗γ , c(ε, α, γ) is given by (6). For α > α∗γ ,
Vol. 66 (2015)
Wave speeds for the FKPP equation
c(ε, α, γ) = cγ0 (α) + c̃(ε, α, γ).
Here,
611
(15)
cγ0 (α)
is the unique solution of
√ 2
q(c, γ)
ω(c, γ)
q(c, γ)
ω(c, γ)
c −4
γ
ln
sin
ln
,
(16)
cos
−
=
c
α2 γ 2
2ω
c
α2 γ 2
q(c, γ)
2
c2 2
(c − 4), and 4(γ + 1) > c2 . Also, c̃
where ω(c, γ) = 12 4(1 + γ) − c2 , q(c, γ) = ω 4 + ω2 (c2 − 2) + 16
satisfies the same properties as the function c̃ in Theorem 1.
Finally, for the FKPP equation (3) with the uniform enhancement function fUE (13), we prove
Theorem 3. For each (α, δ, m) with α ≥ 1, δ > 1, and m = 1, 2, . . ., there exists an ε0 = ε0 (α, δ, m) > 0
such that for ε ∈ (0, ε0 ), the reaction–diffusion equation (3) with enhancement (13) has a unique traveling
wave solution. There also exists a critical value of α given by
1 δ−1 2δ ln 1 −
1+
,
(17)
α∗δ = α∗ (δ) =
δ−1
2
δ
δ
δ
√that for
each δ > 1 there exists an analytic, monotonically increasing diffeomorphism c0 : (α∗ , ∞) →
such
1
2, δ + √δ which is the unique solution of
(1 + δ − cd)d
δ
√
α(c) = √
−1 ,
(18)
d c2 − 4 (dδ)d c2 −4 (d(1 + δ) − c)d2
√
with d = 12 c − c2 − 4 , such that the speeds c(ε, α, δ) of the traveling waves are as follows: For α < α∗δ ,
c(ε, α, δ) is given by (5). For α = α∗δ , c(ε, α, δ) is given by (6). For α > α∗δ ,
c(ε, α, δ) = cδ0 (α) + c̃(ε, α, δ),
(19)
where here c̃ also satisfies the same properties as the function c̃ in Theorem 1.
Remark 2. The ranges of the functions cγ0 (α) and cδ0 (α) obtained with the linear and uniform enhancement
functions fLE and fU E in Theorems 2 and 3 are finite, in contrast to the infinite range of the function
c0 (α) obtained with the constant enhancement function in Theorem 1. The distinction that the speeds
remain finite for linear and uniform enhancement functions is useful for modeling N particle systems and
for lattice models, [36].
The blow-up technique was first used in studying limit cycles in planar vector fields near a cuspidal
loop in [25]. In addition, it has since been applied successfully to a broad array of different systems of
ODEs, including especially in [23] where an extension of the more classical geometric singular perturbation
theory is presented to problems in which normal hyperbolicity is lost; see also [16,18,20,24,32,33,40] and
the references therein. Moreover, the blow-up method has also been used for analyzing propagation speeds
of pulled fronts, bistable fronts, and pushed fronts in continuum reaction–diffusion models with cutoffs,
see [19,21,22,38,39].
This article is organized as follows. In Sect. 2, we employ the geometric desingularization method
(blow-up) to the degenerate equilibrium at Q+ , and we construct the singular heteroclinic orbit. In
Sect. 3, we establish the persistence of this singular heteroclinic for ε > 0 and sufficiently small, thus
completing the proof of Theorem 1. In Sect. 4, we prove Theorems 2 and 3, for the other two primary
enhancement functions studied in this article.
2. Geometric desingularization of (11) and the singular heteroclinics
In this section, we desingularize the origin in system (11) via a blow-up transformation; and, for each
α ≥ 1, we construct a locally unique singular heteroclinic orbit Γ between Q− and Q+ , which corresponds
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F. Dumortier and T. J. Kaper
ZAMP
to the ε = 0 limit of the traveling front solution of (3)–(4). For each fixed value of α ≥ 1, these singular
heteroclinics are independent of m.
The blow-up coordinate change for (11) is
U = r̄ū, V = r̄v̄, and ε = r̄ε̄,
(20)
where (ū, v̄, ε̄) ∈ S2 = (ū, v̄, ε̄) ū2 + v̄ 2 + ε̄2 = 1 , and r̄ ∈ [0, r0 ] for r0 > 0 sufficiently small. With
this coordinate change, the degenerate equilibrium at the origin is transformed into the two-sphere S2 .
Moreover, since we are interested in ε ≥ 0, we only need to consider the half-sphere S2+ defined by
restricting S2 to ε̄ ≥ 0.
As is the case when working with spheres in differential geometry, it is natural to use coordinate
charts here. We analyze the induced vector field on S2+ in the following two charts: the rescaling chart
K2 (ε̄ = 1), which is used to study the dynamics of (11) in Regions 1 and 2, and the phase-directional
chart K1 (ū = 1), which we employ to analyze Region 3, see Sects. 2.1 and 2.2, respectively.
Remark 3. For any object in the original (U, V, ε)-variables, we will denote the corresponding object
in the phase space of the desingularized vector field by . Moreover, in charts Ki (i = 1, 2), the object
will be denoted by i .
2.1. Dynamics in the rescaling chart K2
In this section, we use the rescaling chart K2 , defined by ε̄ = 1 in (20), to study the dynamics of system
(11) in Regions 1 and 2. It provides a top view of the sphere. For each α ≥ 1, we determine the portion
of the singular heteroclinic orbit Γ that lies in K2 .
With ε̄ = 1, the blow-up transformation (20) in this chart is given by
U = r2 u2 ,
V = r2 v2 ,
and
ε = r2 .
(21)
One readily observes that U = ε corresponds to u2 = 1 and U = αε corresponds to u2 = α. Hence,
Region 1 corresponds to 0 ≤ u2 ≤ 1, and in this region, system (11) is equivalent to
u2 = v2 ,
(22a)
v2
r2
= −cv2 ,
(22b)
= 0.
(22c)
The phase space of this linear system is illustrated in Fig. 1. The entire strip 0 ≤ u2 ≤ 1 is foliated with
diagonal lines of slope −c, since the system may be written as dv2 /du2 = −c. Moreover, for each r2 , there
is a line segment of fixed points (u2 , 0, r2 ) parametrized by u2 ∈ [0, 1], and each fixed point on it attracts
the initial conditions on the diagonal line through it.
We focus on the fixed point (0, 0, 0) and denote it by Q+
2 . This fixed point is semi-hyperbolic for (22),
with eigenvalues λ1 = −c, λ2 = 0,, and λ3 = 0. The corresponding eigenspaces are spanned by (1, −c, 0)T ,
(1, 0, 0)T , and (0, 0, 1)T , respectively. The solution of this equation through Q+
2 is
v2 (u2 ) = −cu2 .
(23)
P2∗
= (1, −c, 0); see Figs. 1 and 2. The
This solution hits the right boundary of Region 1 at the point
diagonal line segment between P2∗ and Q+
is
(the
lower
branch
of)
the
stable manifold W2s (Q+
2
2 ), and for
each α ≥ 1, it is precisely the portion Γ2 of the singular solution Γ in Region 1. It will also be useful to
introduce the cross-section Σin
2 = {(1, v2 , r2 )|(v2 , r2 ) ∈ [−v0 , 0] ∪ [0, r0 ]}, where r0 > 0 is sufficiently small
and where v0 is appropriately chosen.
For the parameter values α ≤ α∗ , where we recall that α∗ = 4e2 /(1 + e2 ), we will chose c = 2, while
for α > α∗ , we will chose c = c0 (α) where c0 (α) is the solution of (8). As will become clear below, these
Vol. 66 (2015)
Wave speeds for the FKPP equation
Fig. 1. The phase space of system (11) in Regions 1 and 2, with 1 ≤ α < c2
Fig. 2. The geometry in chart K2
613
614
F. Dumortier and T. J. Kaper
ZAMP
are the limiting values as ε → 0, and for the remainder of this section, we will construct the singular
heteroclinics in terms of c.
Next, we analyze the dynamics of system (11) in Region 2 (1 < u2 ≤ α). In this region, the reaction
term is given by the constant enhancement. The governing equations are
u2 = v2 ,
(24a)
v2
r2
= −cv2 − α,
(24b)
= 0.
(24c)
The phase space of this linear system is the strip 1 < u2 ≤ α, and we may continuously extend solutions
to u2 = 1. The horizontal line v2 = − αc is invariant. Initial conditions in the strip approach it at an
exponential rate in the v2 direction. See Fig. 1.
We focus on orbits that enter the strip at {u2 = α} below the invariant line, i.e., that have initial
conditions (α, v2in , 0) with v2in < − αc , since the orbits with v2in ≥ − αc remain above the line v2 = − αc
and hence cannot connect up to W s (Q+ ) and cannot form the desired heteroclinic. For each orbit with
v2in < − αc , there is an implicit function relating the entry value to the v2 coordinate (v2out ) on exit at
dv2
{u2 = 1}; and, this implicit relation is found by integrating the scalar equation du
= −(cvv22+α) from
2
u2 = 1 to u2 = α:
out α v2 + c
α
v2out − v2in − ln
= c(α − 1).
(25)
c
v2in + αc
Of particular importance is the orbit that exits Region 2 through the point P2∗ = (1, −c, 0), which we
out
recall is the entry point of W2s (Q+
= −c. Hence,
2 ) into Region 1, introduced above. This orbit has v2
in∗
by the implicit function (25), it has v2 coordinate on entry into Region 2 given by v2 , where
−c + αc
α
= −cα.
(26)
v2in∗ + ln
c
v2in∗ + αc
We label the entry point (α, v2in∗ , 0) by P2in∗ , and for each α ≥ 1, this orbit connecting P2in∗ to P2∗ is
precisely that portion of the singular heteroclinic orbit Γ in Region 2.
In summary, for each α ≥ 1, we have identified the portion of the singular heteroclinic orbit Γ in chart
in∗
∗
and Q+
K2 to be that segment of W2s (Q+
2 ) that lies between P2
2 and that goes through the point P2 .
in∗
We label it Γ2 . Also, for reference below, we observe that v2 decreases monotonically with non-zero
speed as c increases.
2.2. Dynamics in the phase-directional chart K1
In this section, we use the directional chart K1 , which is defined by ū = 1, to study the dynamics of
system (11) in Region 3. For each α ≥ 1, we identify the portion of the singular orbit Γ lying in K1 .
With ū = 1, the blow-up transformation in this chart is
U = r1 ,
V = r1 v1 ,
and ε = r1 ε1 ,
(27)
and system (11) is equivalent to
r1 = r1 v1 ,
v1
ε1
= −(1 −
(28a)
r1m )
= −ε1 v1 .
− cv1 −
v12 ,
(28b)
(28c)
For all c, there is a line of fixed points with v1 = 0, −
1 = (1, 0, ε) ε ∈ [0, ε0 ] , which corresponds to the
−
−
original point Q before blowup. In particular, for ε = 0, we will denote the point (1, 0, 0) on −
1 by Q1 .
−
u
The unstable manifold, W1 (Q1 ), is part of the desired singular solution. We will track it in Region 3.
Vol. 66 (2015)
Wave speeds for the FKPP equation
615
We will carry out the tracking separately in the cases c > 2 and c = 2, beginning with the former. For
c > 2, system (28) has two additional equilibria,
c 1 2
c − 4.
(29)
P1± = (0, v1± , 0) where v1± = − ±
2 2
We will focus on P1+ here, and we introduce the notation
d = −v1+ ,
(30)
which will be useful throughout. These fixed points merge at v1 = −1 for c = 2 and then disappear for
c < 2.
√
For c > 2, the eigenvalues of (28) linearized at P1+ are given by −d, − c2 − 4, and d, with eigenvectors
(1, 0, 0)T , (0, 1, 0)T , and (0, 0, 1)T , respectively. Hence, P1+ has a two-dimensional stable manifold given by
{ε1 = 0}, and a one-dimensional unstable manifold corresponding to the invariant line v1 = −d, r1 = 0.
The key observation is that, in the plane ε1 = 0, solutions along the unstable manifold W1u (Q−
1 ) approach
P1+ with v1 → v1+ from above, and we remark that the details of the approach depend on the size of c,
as there is a 1 : 1 resonance among the two stable eigenvalues at c = √32 .
+
+
Lemma 2.1. Orbits in the unstable manifold W1u (Q−
1 ) approach the fixed point P1 with v1 < v1 , i.e.,
3
from above, and this approach is tangent to the v1 -axis for c0 ∈ (2, √2 ) and perpendicular to the axis for
c0 > √32 .
Proof. The proof of this lemma follows from the same type of phase plane argument used in the proof of
Lemma 2.5 in [21]. Consider system (11) in the limit ε = 0, in which it is the traveling wave ODE system
for the original FKPP equation. We construct a trapping region in the (U, V ) phase plane and show that
orbits on W u (Q− ) must enter it and approach Q+ inside it.
The fixed point Q+ is a stable node for c > 2 and a stable improper node for c = 2. The upper boundary
of the trapping region is given by the U -axis, i.e., {V = 0}. On this axis, the vector field simplifies to
U = 0 and V = −U (1 − U m ). Hence, the vector field points downward there for√U ∈ (0, 1), into the
1
2
trapping region. The lower boundary of the trapping region is the line V = ( −c
2 + 2 c − 4)U . This line
+
corresponds
to the weak stable eigendirection at Q , corresponding to the weak stable eigenvalue λ+ =
√
−c
1
2 − 4 ≡ −d. Projecting the vector field (11) onto this eigendirection, we find U (1−U m )+cV +dV .
+
c
2
2
Now, on V = −dU , the sign of this expression is such that, for all U ∈ [0, 1] and for both c > 2 and
c = 2, the vector field points into the trapping region at all points on the lower boundary. Putting these
two results together, we see that orbits on W u (Q− ) must enter the trapping region and hence approach
Q+ . Finally, from the governing equations in chart K1 , one sees that the two stable eigenvalues, −d and
2d − c, are equal at c = √32 . Therefore, since the approach to P1+ is along the weak stable direction, the
approach is either tangent to the v1 -axis or perpendicular to it, depending on which side of √32 the speed
c lies on. This completes the proof of the lemma.
For all α ≥ 1, the first portion of the singular solution in chart K1 is then precisely W1u (Q−
1 ), and we
.
label it by Γ−
1
The second portion, denoted Γ+
1 , of the singular orbit Γ1 depends on the magnitude of α relative to
α∗ . For α > α∗ , the second portion of the singular orbit Γ1 is given by the segment of the one-dimensional
manifold W1u (P1+ ), which coincides with {v1 = −d, r1 = 0}, up to the point P1out = (0, −d, α1 ). Then, for
α = α∗ , it is given by the segment of the one-dimensional unstable manifold of (0, −1, 0) up to the point
P1out = (0, −1, α1 ). Finally, for α < α∗ , it corresponds to the trajectory that leaves (0, −1, 0) tangent to
the v1 -axis with v1 < −1 and that hits the plane ε1 = α1 at the point P1out = (0, v1out , α1 ), with v1out < −1.
The illustration in Fig. 3 is for the case α > α∗ . where the cross-section Σin
1 = {(r0 , v1 , ε1 )|(v1 , ε1 ) ∈
[−v0 , 0] ∪ [0, 1]}, where v0 is the same as in the previous section. Also, for reference below, we observe
that, for each α ≥ 1, v1out is strictly monotonically increasing as c increases.
616
F. Dumortier and T. J. Kaper
ZAMP
Fig. 3. The geometry in chart K1 for c > 2 and α > α∗ The illustration is for the case 2 < c <
3
√
2
2.3. Completing the construction of the singular heteroclinic Γ
The final step in the construction of the singular orbit for each α ≥ 1 is to identify the exit point P1out ,
where it leaves Region 3 and chart K1 , with the point P2in∗ , where it enters Region 2 and chart K2 . With
+
this identification, we will have hooked up the pieces, Γ−
1 , Γ1 , and Γ2 , of the singular orbit and connected
−
+
Q1 to Q2 in {r2 = 0}. Here, we present the constructions for α > α∗ and α = α∗ . In the third (and
remaining) case α < α∗ , we chose the unique orbit inside {r1 = 0} for chart K1 that cuts the plane {ε1 =
1
in∗
α } at the point P2 . We refer the reader to Section 2.3 of [21] for the details of the construction for α < α∗ .
To relate the analyses of the previous sections, we use the following relationship between the variables
in (21) and (27) on the domain of overlap between charts K1 and K2 :
Lemma 2.2. [21] The change of coordinates κ12 : K1 → K2 is given by
1
v1
u2 = , v2 = , and r2 = r1 ε1 .
ε1
ε1
For the inverse change κ21 = κ−1
12 : K2 → K1 , there holds
v2
, and
r1 = r2 u2 , v1 =
u2
ε1 =
1
.
u2
Both κ12 and κ21 are well defined as long as ε1 and u2 , respectively, are finite and bounded away from
zero. Correspondingly, the overlap domain between K1 and K2 includes {U = αε}, where ε1 = α1 and
u2 = α.
Recall P1out = (0, −d, α1 ) and P2in∗ = (α, v2in∗ , 0). Hence, using the lemma, we find that v2in∗ =
out out
v1 /ε1 = −αd. Therefore, after substitution of this into (26) and some algebra, we find the desired
relation between c and α:
Vol. 66 (2015)
Wave speeds for the FKPP equation
617
Fig. 4. The global geometry of the blown-up vector field
(c2 − α)ec(c−d) + α − αcd = 0,
(31)
where c > 2 for d < 1 and α > α∗ . Solving this for α as a function of c, we find (8). In addition, in the
4e2
limit c → 2+ , we see from (31) that α → α∗ , where α∗ = 1+e
2 , as stated in Theorem 1. Moreover, these
results may be extended continuously to v1 = −1 in the case in which c = 2 and α = α∗ .
0
Finally, one may verify that dα
dc (c0 ) > 0. A direct calculation yields
√
√
dα0
(α + eP (c) (c2 − α))(c2 − 2 − c c2 − 4) + 2eP (c) c c2 − 4(c2 − α + 1)
√
=
,
dc
eP (c) − 1 + 2c (c − c2 − 4)
√
where P (c) = 2c (c + c2 − 4) and 1 < α∗ ≤ α < c2 < eP (c) , so that the numerator and denominator are
strictly positive. Alternatively, one may see this from the observations that the values v1+ (= −d) and v2in∗
are monotonically increasing and decreasing, respectively, as c increases, so that the manifolds W u (Q− )
and W s (Q+ ) pass through each other transversely at c = c0 (α).
Therefore, we have completed the construction of the desired singular heteroclinic orbit for each
parameter α ≥ 1. This construction is summarized in the following proposition, and global geometry of
the singular heteroclinic solution is illustrated in Fig. 4.
Proposition 2.1. For each (α, m) with α ≥ 1 and m = 1, 2, . . ., there exists a singular heteroclinic orbit
+
−
+
Γ that connects Q−
1 to Q2 and that consists of the union of the segments Γ1 , Γ1 , and Γ2 of equations
2
4e
(22), (24), and (28). There exists α∗ = 1+e
2 such that for each α ≤ α∗ , c0 (α) = 2, whereas for each
α > α∗ , c is given by the unique solution c0 (α) of the equation α = α0 (c), where α0 is given by (8).
0
Moreover, c0 (α) > 2 for all α > α∗ , dα
dc (c) > 0 for c > 2, and at c = c0 (α), the singular orbit lies in the
u
−
transverse intersection of W (Q ) and W s (Q+ ) for ε = 0. Finally, in chart K1 for c > 2, the singular
618
F. Dumortier and T. J. Kaper
heteroclinic passes through P1+ = (0, −d, 0), where d =
(0, −1, 0).
c
2
−
1
2
ZAMP
√
c2 − 4, and for c = 2, it passes through
Remark 4. The geometry in Region 2 gives rise naturally to the existence of the critical value α∗ stated
in the proposition. For each value α > α∗ , we see that v2in∗ = −αd < −α, and there is a unique value of
c > 2, for which a singular heteroclinic connection can be formed involving the unstable manifold of the
saddle at v1 = v1+ = −d, as stated in the proposition. At α = α∗ , v2in∗ = −α, and the singular heteroclinic
connection can only be formed for c = 2. Finally, for values α < α∗ , the situation is similar to that in
which there is no enhancement.
3. Persistence of the singular heteroclinic orbit Γ and completion of the Proof of Theorem 1
In this section, we establish the persistence of the singular solution for each (α, m) with α ≥ 1 and
m = 1, 2, . . . and, hence, complete the proof of Theorem 1. We start in Sect. 3.1 by showing that, for each
such (α, m) and for each ε > 0 sufficiently small, there exists a unique value c(ε, α) of c in (11), for which
there is a heteroclinic orbit connecting Q− to Q+ that lies in the intersection of W u (Q− ) and W s (Q+ ),
and that is close to the singular orbit Γ constructed in the previous section. Then, in Sect. 3.2, we derive
the corresponding necessary conditions involving the dependence of c̃(ε, α) on ε in order for Γ to persist.
Combining these two aspects of the analysis, we obtain the existence of the unique traveling waves with
wave speeds c(ε, α), as stated in Theorem 1. We carry out the construction explicitly first for the case
in which c0 (α) > 2 and α > α∗ and then for the case in which c0 (α) = 2 and α = α∗ . The construction
for c0 (α) = 2 and α ∈ [1, α∗ ) is similar to that given in Section 3 of [21] for the FKPP equation without
an enhancement. For convenience of notation, we suppress the dependence of the invariant manifolds
W u (Q− ) and W s (Q+ ) on the system parameters in (11), including c, as is customary in dynamical
systems theory.
3.1. Existence and uniqueness of persistent heteroclinics
In this section, we establish the existence and uniqueness of the persistent heteroclinics with c(ε, α) ∼
c0 (α) for each (α, m) with α ≥ 1. We prove that, for each (α, m) with α ≥ 1 and m = 1, 2 . . . and for
ε > 0 sufficiently small in (11), the unstable manifold of Q− intersects the stable manifold of Q+ for a
unique value of c, labeled c(ε, α).
Proposition 3.1. For each (α, m) with α ≥ 1 and m = 1, 2, . . ., there exists an ε0 sufficiently small such
that for 0 < ε < ε0 and c(α, ε) ∼ c0 (α), there exists a unique heteroclinic connecting Q− to Q+ .
Proof. The proof of this proposition is similar to that of Proposition 3.1 in [21]. In the system with ε = 0
(i.e., r1 , r2 = 0) and c = c0 (α), the intersection of W u (Q− ) and W s (Q+ ) is transverse for each α ≥ 1, as
shown in the previous section. There, it was shown that −d, the v1 coordinate of the point P1out at which
out
in∗
W1u (Q−
1 ) intersects Σ1 , is strictly monotonically increasing as the parameter c increases, whereas v2 ,
+
in∗
s
in
out
the v2 coordinate of the point P2 at which W2 (Q2 ) intersects Σ2 = κ12 (Σ1 ), is strictly monotonically
decreasing as the parameter c increases. Hence, the manifolds pass through each other with non-zero
speed at c = c0 (α).
Then, by standard persistence theory for invariant manifolds under smooth perturbations, the full
system with ε positive and sufficiently small has a smooth stable manifold W s (Q+ ) on {0 ≤ U < ε}, which
may be extended smoothly to U = ε. Similarly, by standard persistence theory for invariant manifolds
under smooth perturbations, the full system with ε positive and sufficiently small has a smooth unstable
manifold W u (Q− ) on the domain αε < U ≤ 1, and it may be extended smoothly to U = αε. In addition,
solutions on the unstable manifold W u (Q− ) may be extended continuously across the discontinuity at
Vol. 66 (2015)
Wave speeds for the FKPP equation
619
U = αε along smooth solutions of the full problem on the domain ε ≤ U < αε. These manifolds are O(ε)
close to their ε = 0 counterparts, and hence, due to the transverse intersection of their ε = 0 counterparts,
these perturbed manifolds intersect transversally for a unique value of c(ε, α), where c → c0 (α) as
ε → 0.
3.2. Transition through chart K1 and the asymptotics of c̃(ε, α)
In this section, we study the passage of trajectories through chart K1 under the flow of the full equation (28) with r1 > 0, and we derive the asymptotics of c̃.
It is convenient to work with appropriate sections for the flow. We will employ the sections Σout
=
1
in
in
out
is governed
κ21 (Σin
2 ) and Σ1 defined above. The transition of orbits through chart K1 from Σ1 to Σ1
out
by the transition map Π1 : Σin
1 → Σ1 . We will derive asymptotically accurate representations of this
map in both cases, α > α∗ and α = α∗ , beginning with the former. The analysis of the transition through
K1 for r1 > 0 in the third case, α < α∗ , follows closely that of Section 3.2 in [21] for the FKPP equation
without enhancement.
We begin with the case α > α∗ . Taking into account that c(ε, α) → c0 (α) for each α > α∗ in the
singular limit as ε → 0, we define c̃(ε, α) = c(ε, α) − c0 (α), where we observe that c̃(ε, α) = O(1) as ε → 0
by Proposition 3.1. We shift the point P1 = (0, v1+ = −d, 0) to the origin by introducing the new variable
w = v1 + d = v1 − v1+ .
(32)
in
As before, let P1in denote the point of intersection of Γ−
1 with Σ1 . Since c̃ ∼ 0, we restrict ourselves to
+
+
in
describing Π1 on Σ1 ∩ {v1 > v1 } in the following.
With the above transformations, system (28) is equivalent to
r1 = −r1 (d − w),
r1m )
ε1
= ε1 (d − w).
w = −(1 −
(33a)
2
+ c(d − w) − (d − w) ,
(33b)
(33c)
Then, recalling that 1−cd+d2 = 0 and rescaling time by the positive factor d−w, we find that system (33)
is equivalent to
ṙ1 = −r1 ,
ẇ =
(2 −
(34a)
+ d1 (r1m
1 − wd
c
d )w
2
−w )
ε̇1 = ε1 .
,
(34b)
(34c)
(Here, the overdot denotes differentiation with respect to the new, rescaled time ξ1 .) Plainly, the ε1 equation decouples, and ε1 (ξ1 ) = ε1,0 eξ1 = rε0 eξ1 . Also, r1 (ξ1 ) = r0 e−ξ1 , but it will be useful to keep the
equations for r1 and w together as a system.
To study the (r1 , w)-system in (34), we binomially expand the second component for wd < 1 and put
the system into normal form.
Lemma 3.1. There exists a smooth near-identity coordinate change of the form (r1 , w) → (r1 , W ) with
W = wk(r1 , w), with k(0, 0) = 1, which transforms (34) into
ṙ1 = −r1 ,
c
rm
Ẇ = 2 −
W+ 1 .
d
d
(35a)
(35b)
620
F. Dumortier and T. J. Kaper
ZAMP
Proof. Let ν = dc − 2. We have ν > 0. At the level of formal power series, a calculation with Lie brackets,
following the general presentation [13,27], yields
∂
∂ ∂
∂
, wi r1j
.
− νw
− r1
= (−j − (i − 1)ν)wi r1j
∂r1
∂w
∂w
∂w
∂
Hence, it clearly follows that all terms of the form wi r1j ∂w
with i ≥ 1 can be removed by a transformation
w → W1 = wh̃(r1 , w) for some formal power series h̃.
Now, by taking any smooth function h(r1 , w) with infinite jet j∞ h(0, 0) = h̃, one has that the transformation (r1 , w) → (r1 , W1 ) with W1 = wh(r1 , w) brings (34) into the form
ṙ1 = −r1 ,
(36a)
m
c
r
Ẇ1 = 2 −
W1 + 1 + W1 (r1 , W1 ),
(36b)
d
d
where is a smooth function whose formal power series expanded about the point (r1 , W1 ) = (0, 0) is
identically zero.
Finally, from the normal form theory presented in [41], the flat function in the second component
of the system may also be removed by a near-identity coordinate change that preserves {W1 = 0}.
Hence, (36) has been transformed into system (35) via a sequence of smooth, near-identity coordinate
changes that preserve {w = 0}.
Next, we solve the normal form (35) for arbitrary α > α∗ . For convenience, we set c̃(α, ε) = −η in the
decomposition c(α, ε) = c0 (α) + c̃(α, ε), suppressing the α-dependence, where η is a new variable. The
initial conditions are r0 (0) and W (0) = w0 + b(η). Using explicit integration, we find
r1 (ξ1 ) = r1 (0)e−ξ1
(r1 (0))m (2− c )ξ1
(r1 (0))m
e−mξ1 .
e d −
W (ξ1 ) = w0 + b(η) +
(m + 2)d − c
(m + 2)d − c
(37a)
(37b)
Then, from the solution ε1 (ξ1 ), we know that e−ξ1 = r1εα
(0) , where ξ1 denotes the time at which the
out
solutions hit the section Σ1 at {u2 = α}. Therefore, the solutions are given by
(r1 (0))m εα dc −2
α m εm
.
−
W (ε, η) = w0 + b(η) +
(m + 2)d − c r1 (0)
(m + 2)d − c
√
With the expansion c = c0 − η, we also have d = d0 + δ(η), where d0 = c20 − 12 c2 − 4 and δ (0) < 0,
as well as dc − 2 = dc00 − 2 + τ (η), with τ (0) = 0. Therefore, on the cross-section Σout
at the boundary
1
between charts K1 and K2 , the final form of the solution for each (α, m) is
εα dc0 −2+τ (η)
(r1 (0))m
α m εm
0
(38)
−
W (ε, η) = w0 + b(η) +
(m + 2)d0 − c0 + δ̃(η) r1 (0)
(m + 2)d0 − c0 + δ̃(η)
For each (α, m), this solution, which lies on W u (Q− ) and which we tracked through Region 3 in chart
K1 , must also lie on W s (Q+ ) to represent the persistent heteroclinic. This is the geometric condition
that uniquely determines η (i.e., −c̃) as a function of ε and α.
To impose this geometric condition, we observe that for small positive values of ε = r2 > 0, the
manifold W s (Q+ ) may also be tracked in chart K2 , just as we did in Sect. 2.1 in the limiting case
ε = r2 = 0. The equations in chart K2 depend on ε only through the value of c = c0 (α) − η. Hence, by
following the calculations presented in Sect. 2.1, we see that orbits on W2s (Q+
2 ) hit the line u2 = 1 at
the boundary of the cutoff domain with v2 = −(c0 (α) − η). Moreover, we can also follow the calculations
presented there for Region 2. In particular, using formula (25) with v2out = −(c0 (α)−η), one can calculate
the corresponding value of v2 , denoted v2in∗ (η), at which the orbits on W2s (Q+
2 ) cross the line u2 = α,
out
=
κ
(Σ
).
From
this
calculation,
one
also
sees
that
v2in∗ (η) approaches v2in∗
i.e., hit the section Σin
12
2
1
Vol. 66 (2015)
Wave speeds for the FKPP equation
621
from below and at a linear rate in the limit η → 0, where v2in∗ is given by (26). Namely, we may write
v2in∗ (η) = v2in∗ − ηs(η) for some bounded function s satisfying s(0) < 0.
Hence, imposing the geometric condition and using only the leading-order terms in (38), we find that
to leading order
εα d0 −2
(r1 (0))m
α m εm
η
0
−
= − s(0).
(m + 2)d0 − c0 r1 (0)
(m + 2)d0 − c0
α
c
w0 +
(39)
For each α > α∗ , this formula determines how η and, hence, c̃ depend on ε, and we recall that w0 depends
on r1 (0).
We demonstrate this as follows. The term on the left-hand side that dominates, asymptotically as
ε → 0, depends on the value of c. In particular, we see that the exponent dc00 − 2 on ε in the first term
equals the exponent m of the second term when c = √m+2
, for each m = 1, 2, . . .. These are precisely the
m+1
, we see that dc00 − 2 < m, which means
values at which there are resonances in (34). Hence, for c < √m+2
m+1
that the first term dominates, and
c0
c0
w0
−α d0 −1
d0 −2
lim
η∼
ε
.
c0
s(0)
r1 (0)→0 (r (0)) d0 −2
1
In contrast, for c >
√m+2 ,
m+1
we see that
c0
d0
η∼
− 2 > m, which means that the second term dominates, and
αm+1
s(0)((m + 2)d0 − c0 )
Summarizing, formula (39) reveals that
⎧
c0
⎪
O(ε d0 −2 )
⎪
⎪
⎨
η = O(εm ln(1/ε))
⎪
⎪
⎪
⎩ O(εm )
εm
if c <
√m+2 ,
m+1
if c =
√m+2 ,
m+1
if c >
√m+2 .
m+1
(40)
In the boundary case, the gauge function ln(ε) arises due to the resonance. The coefficient may be
computed directly from (38).
In all cases, we have that η > 0 (where we recall s(0) < 0), and hence, c̃(ε,
the
α) =√−η < 0. Also,
1
2
2
coefficients will depend on α. Finally, we recall that β0 (c) was defined to be 2 c + c c − 4 − 4 in (9),
and this is precisely dc00 − 2. Therefore, asymptotically as ε → 0, formula (40) exactly yields the function
for c̃(ε, α) = −η stated in (10). This completes the proof of Theorem 1 for each (α, m) with α > α∗ and
m = 1, 2, . . ..
4e2
Now, we prove Theorem 1 in the case α = α∗ = 1+e
2 . For this case, c0 (α) = 2, and the analysis
proceeds closely along the lines used in Section 3.2 of [21] in the case of the FKPP equation without
enhancement, although it differs in one crucial aspect. Following the analysis of [21], we write c(α, ε) =
2 − η 2 , where we make the Ansatz that the correction to the wave speed is negative. This Ansatz is made
without loss of generality, since the wave speed is unique and we find a solution with c̃ < 0.
Since v1+ = −1, system (33) is equivalent in this case to
r1 = −r1 (1 − w),
r1m )
ε1
= ε1 (1 − w).
w = −(1 −
(41a)
2
2
+ (2 − η )(1 − w) − (1 − w) ,
(41b)
(41c)
622
F. Dumortier and T. J. Kaper
ZAMP
Hence, after simplifying the middle component and rescaling the time in the full system by the positive
factor 1 − w, we arrive at the main system to be studied in this case,
ṙ1 = −r1 ,
ẇ = −η 2 +
(42a)
r1m − w2
,
1−w
η̇ = 0.
(42b)
(42c)
This is a system of two first-order equations which depend on the parameter η.
To analyze the transition through chart K1 as the heteroclinics pass near P1 for ε ∈ (0, ε0 ) small, we
compute the map Π1 in more detail. The functional relation between η and ε will follow from requiring
out
that the unstable manifold W u (Q− ) connects to the stable manifold W s (Q+ ) in Σin
2 = κ12 (Σ1 ) after
the transition past P1 .
Proposition 3.2. Let α = α∗ and c0 (α) = 2. For a heteroclinic connection between Q− and Q+ to be
possible when ε > 0 in (11), η must be given by
π
+ O (ln ε)−2 .
(43)
η(ε) = −
2 ln ε
Proof. To simplify the analysis of (42), we make a normal form transformation which decouples the
dynamics of r1 and w in (42). For each r ≥ 1, there exists, by Theorem 1 of [7], a C r coordinate change
(44)
(r1 , w) → R1 (r1 , w, η), W (r1 , w, η)
with R1 (0, w, η) = 0 which transforms the r1 − w system (42) into
Ṙ1 = −R1 ,
(45a)
2
W
,.
(45b)
1−W
This normal form with η as a parameter respects the invariance of {r1 = 0}.
1 of solutions of system (45) between the
For η sufficiently small, we calculate the transition “time” Ξ
out
and
Σ
after
transformation
by (44). Let W in > 0 and W out < 0
two sections corresponding to Σin
1
1
1 (W in , W out , η) is
1 = Ξ
denote the corresponding values of W . We will see that, to leading order, Ξ
in
out
independent of the exact values of W and W .
Since the equations in (45) are decoupled, we can solve (45b) by separation of variables. Introducing
2
the new variable Z = W − η2 in (45b), we find
2
1 − Z − η2 dZ
.
−dξ˜1 =
2
Z 2 + η 2 1 − η4
Ẇ = −η 2 −
Integrating, we find
⎛
η2
1
−
1 = 2 arctan ⎝ Z
−Ξ
2
η 1 − η4
η 1−
⎞
Z out
Z out
1 2
η 2 2
⎠
in − 2 ln Z + η 1 − 4 in .
η2
Z
Z
(46)
4
Here, Z in and Z out are the values of Z obtained from W in and W out , respectively.
Reverting to W in (46) and dividing out a factor of η −1 , we find
⎡
⎛
⎛
⎡
⎞
⎞⎤
η2
η2
η2
out
in
− 2
1 = 1 ⎣ 1 − 2 ⎣arctan ⎝ W − 2 ⎠ − arctan ⎝ W
⎠⎦
−Ξ
2
η
η2
η2
η 1− 4
η 1 − η4
1− 4
"
η! − ln (W out )2 − W out η 2 + η 2 − ln (W in )2 − W in η 2 + η 2 .
2
(47)
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Wave speeds for the FKPP equation
623
2
2
1
Since we are only interested in deriving a leading-order expression for η, we expand (1 − η2 )(1 − η4 )− 2 =
1 + O(η 2 ). Also, since W out < 0 and W in > 0 and since (44) is near identity, we conclude that W out < 0
and W in > 0 are O(1) as ε → 0 and independent of η to leading order.
To derive expansions for the arctangent terms in (47), we make use of the identity
1
π
=± ,
arctan(x) + arctan
x
2
where the sign equals the sign of x. In particular, for |x| large, we have [1]
1
π 1
− 3 + ... .
arctan(x) = ± −
2
x 3x
In our case,
2
W − η2
W
1 + O(η 2 )
=
x= 2
η
η 1 − η4
and, hence,
⎞
2
W out − η2
⎠ = − η + O(η 3 )
arctan ⎝ W out
η2
η 1− 4
⎞
2
W in − η2
⎠ = π − η + O(η 3 ).
arctan ⎝ 2
W in
η2
η 1− 4
⎛
⎛
and
The crucial difference lies with this first arctan, which is zero to leading order here due to the fact that
Γ+
1 lies on the line defined by w = 0 and r1 = 0 perpendicular to the w-axis, whereas that arctan term
is asymptotic to − π2 for α < α∗ due to the fact that Γ+
1 is then tangent to the w-axis.
For the logarithmic terms in (47), one has ln |W 2 − W η 2 + η 2 | = ln |W 2 | + ln |1 −
2 ln |W | + O(η 2 ). Hence,
out W 1
1
1 = 1 π + η
+ O(η 3 ) .
Ξ
−
+
ln
W in η 2
W out
W in
η2
W 2 (1
− W )| =
(48)
On the other hand, we know that R1 evolves according to R1 = R1in e−ξ̃1 , where R1in > 0 denotes the
is also given by
initial value R1 (0). Hence, the time Ξ
Ξ̃1 = − ln
R1
= − ln r1 β(r1 , w, η) ,
in
R1
(49)
where β(r1 , w, η) is a strictly positive, C r -smooth function that depends on the choice of normalizing
coordinates in (44). (Note that the ε-dependence of β is implicitly encoded in its arguments r1 , w,
1 introduced above, the R1 and η and that β = O(1) as ε → 0.) Now, during that same “time” Ξ
in
out
s +
variable has to evolve from R1 to a value R1 permitting a connection between W1u (−
1 ) and W2 (2 ) in
in
out
Σ2 = κ12 (Σ1 ). This will impose a relation between η and ε that we calculate to leading order in the
is fixed at r1out = ε. Hence, (49)
normal form coordinates (R1 , W, η). To that end, recall that r1 in Σout
1
yields
1 = − ln εβ̃(ε, η)
Ξ
(50)
for some function β̃(ε, η) which is strictly positive and C r -smooth, with β̃ = O(1) for ε → 0.
Combining (48) and (50) and recalling that W in and W out are O(1) as ε → 0, as noted below Eq. (47),
we find
1 π
+ ηθ(ε) + O η 2
(51)
− ln ε =
η 2
624
F. Dumortier and T. J. Kaper
ZAMP
for some bounded function θ. Solving (51) for η, we obtain
π
+ η̃,
(52)
η=−
2 ln ε
where η̃ defines a relative correction in (52), i.e., there holds η̃ = O((ln ε)−1 ). In fact, substituting (52)
into (48), one can check that η̃ = O((ln ε)−2 ): Given (51), it follows that
π
− η̃ ln ε θ(ε) + O (ln ε)−1 , η̃, η̃ 2 ln ε .
η̃(ln ε)2 =
2
Since η̃ ln ε = O(1) by assumption, we have η̃ = O((ln ε)−2 ). This concludes the proof.
Now, we show that the assertions of Theorem 1 for α = α∗ follow immediately from Propositions 3.1
and 3.2. By Proposition 3.1, given ε > 0 sufficiently small, for α = α∗ and for each m = 1, 2, . . ., there
exists a heteroclinic orbit in (11) connecting Q− and Q+ and each heteroclinic lies close to the correπ2
sponding singular heteroclinic orbit Γ with c ∼ 2 − 4(ln(ε))
2 . Then, by Proposition 3.2, the heteroclinics
lie in the intersection of the two manifolds W u (Q− ) and W s (Q+ ) and correspond to the traveling wave
solutions of (3) with enhancement (4). This establishes (6) and completes the proof of Theorem 1 also
for the case α = α∗ .
4. Proofs of Theorems 2 and 3
In this section, we present the proofs of Theorems 2 and 3, beginning with the former.
Proof. For the PDE (3) with reaction function f given by (12), the traveling wave ODE is again (11),
and the blow-up coordinate change is again given by (20). Moreover, the function fLE is the same as the
constant enhanced reaction function (4) in both Region 1 (0 ≤ U ≤ ε) and Region 3 (αε < U ≤ 1). Hence,
in Region 1, the desingularized vector field in chart K2 is equivalent to (22), and the desired solution is
again (23). It enters Region 1 at the point P2∗ = (1, −c, 0). It is a segment of W s (Q+
2 ) and constitutes
the portion of the singular heteroclinic orbit Γ2 in Region 1, exactly as in Sect. 2.1.
Similarly, the governing system of equations in Region 3 is equivalent to (28), because fLE is also
the unperturbed reaction function here. Moreover, all of the analysis in Sect. 2.2 applies directly. The
−
+
Γ−
portions of the singular heteroclinic in Region 3 are given by
1 is the connection from
√ Γ1 ∪ Γ1 , where
−
+
+
+
+
c
1
2
Q1 = (1, 0, 0) to P1 = (0, v1 , 0) with v1 = −d = − 2 + 2 c − 4, and Γ1 is that part of W u (P1+ ) on
the line {r1 = 0, v1 = −d} up to the point P1out = (0, −d, 1/α).
The governing system of equations in Region 2 (1 < u2 ≤ α) differs from that generated by the
constant enhancement function (4). In Region 2, in terms of the variables of chart K2 , the governing
equations are
u2 = v2 ,
(53a)
v2
r2
= −cv2 − (1 + γ)u2 ,
(53b)
= 0.
(53c)
This is the equation for a damped harmonic oscillator.
To find the portion of the singular heteroclinic orbit Γ in Region 2, we look for a solution of (53) that
connects the point P1out = (r1 = 0, v1 = −d, ε1 = 1/α), at which the singular heteroclinic exits Region 3
and enters Region 2, to the point P2∗ = (u2 = 1, v2 = −c, r2 = 0), at which the singular heteroclinic exits
Region 2 and enters Region 1. Also, we recall that under the coordinate change between charts K1 and
K2 , P1out is the same as P2in∗ = (u2 = α, v2 = −dα, r2 = 0).
For 4(γ + 1) > c2 , the general solution of (53) is
u2 (ξ) = Ae−δ̃ξ cos(ωξ) + Be−δ̃ξ sin(ωξ).
(54)
Vol. 66 (2015)
Wave speeds for the FKPP equation
625
Straightforward substitution of this solution into the equation yields
1
c
and ω =
δ̃ =
4(1 + γ) − c2 .
(55)
2
2
The equation is autonomous. Hence, without loss of generality, we may set ξ = 0 as the time of entry into
Region 2, and we denote the time of exit from Region 2 by ξexit . The desired solution satisfies u2 (0) = α,
v2 (0) = −dα, u2 (ξexit ) = 1, and v2 (ξexit ) = −c. Applying all four of these conditions, we find
A = α,
α 2
α
c − 4.
B = (δ̃ − d) =
ω
2ω
(56a)
(56b)
Also,
Ae−δ̃ξexit cos(ωξexit ) + Be−δ̃ξexit sin(ωξexit ) = 1,
(−δ̃A + Bω)e−δ̃ξexit cos(ωξexit ) − (Aω + B δ̃)e−δ̃ξexit sin(ωξexit ) = −c,
(57a)
(57b)
where δ̃ and ω are given by (55). Solving this linear system of two equations in two unknowns, we find
1 δ̃ξexit 2 c 2
e
cos(ωξexit ) =
(ω −
c − 4),
(58a)
αγ
4
ω δ̃ξexit
e
sin(ωξexit ) =
(c + c2 − 4).
(58b)
2αγ
Hence, using the fundamental trigonometric identity, we find
q(γ, c)
1
ξexit = − ln
,
c
α2 γ 2
2
(59)
2
c
(c2 − 4). Finally, substitution of this result into equation (56a) yields
where q(γ, c) = ω 4 + ω2 (c2 − 2) + 16
the desired relation (16) between c and α (and γ).
√
Moreover, the requirement that 4(γ + 1) > c2 is equivalent to c < 2 γ + 1, and this is the√ upper
bound on the values of c for which the desired heteroclinic solution exists. In fact, for each c ≥ 2 γ + 1,
the system has an invariant line. The point P2in∗ = (α, dα), which is the entry point of solutions on
W u (Q− ) into Region 2, lies above this line, whereas the point P2∗ = (1, −c), which is the point at which
s
+
solutions
√ on W (Q ) exit Region 2 and enter into Region 1, lies below the invariant line. Hence, for each
c ≥ 2 γ + 1, it is not possible to construct this type of singular heteroclinic solution. Also, one may
show that dα
dc > 0 by applying phase plane analysis, as we show in a more detailed manner in the proof
of Theorem 3 below. This provides complete information about the desired singular heteroclinic orbit in
Region 2.
Next, we calculate the critical value of the parameter α, denoted α∗γ . This critical value is obtained in
√
the limit c → 2+ , in which also d → 1− , δ̃ → 1+ , and ω → γ − . Hence, B = 0, and the solution simplifies
to
√
(60a)
u∗2 (ξ) = α∗γ e−ξ cos( γξ),
√
√
∗
γ −ξ
γ √ −ξ
(60b)
v2 (ξ) = −α∗ e cos( γξ) − α∗ γe sin( γξ).
Therefore, from (57), one readily finds
1
ξexit = √ arctan
γ
and in turn, one obtains
α∗γ
ξexit
=e
√
sec( γξexit ) =
1
√
γ
,
1
1
γ+1
exp √ arctan √
,
γ
γ
γ
(61)
(62)
626
F. Dumortier and T. J. Kaper
ZAMP
exactly as stated in formula (14) in Theorem 2. Therefore, for α ≤ α∗γ , cγ0 ∼ 2 and for α > α∗γ , cγ0 is given
by an analytic, monotonically increasing diffeomorphism.
Finally, we observe that the trapping region argument, the proof of the persistence of the singular
heteroclinic orbit for 0 < ε 1, and the rigorous asymptotics for c̃ given in Sect. 3 carry over to this
system, as well, with appropriate modifications. This completes the proof of Theorem 2.
To conclude this section, we present the essential steps in the proof of Theorem 3.
Proof. For the PDE (3) with reaction function f given by (13), the traveling wave ODE is again (11),
and the function fUE is the same as the constant enhanced reaction function (4) in both Regions 1 and 3.
Hence, all of the analysis in Sects. 2.1 and 2.2 for Regions 1 and 3 applies directly. It remains to analyze
the dynamics in Region 2, just as in the previous proof.
In Region 2 (1 < u2 ≤ α), the governing equations are
u2 = v2 ,
(63a)
v2
r2
= −cv2 − (u2 + δ),
(63b)
= 0,
(63c)
in terms of the variables of chart K2 , with δ > 1. This is the equation for a damped harmonic oscillator
with constant forcing. The equilibrium point is located at (−δ, 0).
To find the portion of the singular heteroclinic orbit Γ in Region 2, we look for the solution of (63) that
connects the point P1out = (r1 = 0, v1 = −d, ε1 = 1/α), at which the singular heteroclinic exits Region 3
and enters Region 2, to the point P2∗ = (u2 = 1, v2 = −c, r2 = 0), at which the singular heteroclinic exits
Region 2 and enters Region 1. Again, we recall that under the coordinate change between charts K1 and
K2 , P1out is the same as P2in∗ = (u2 = α, v2 = −dα, r2 = 0).
First, we solve system (63) in the critical case c = 2 (and hence, d = 1), in which there is a double
root of the indicial polynomial:
(64)
u∗2 (ξ) = Ae−ξ + Bξe−ξ − δ.
Here, we also set ξ = 0 as the time of entry into Region 2, and we denote the time of exit from Region
2 by ξexit . Hence, the desired solution satisfies u∗2 (0) = α, v2∗ (0) = −α, u∗2 (ξexit ) = 1, and v2∗ (ξexit ) = −2.
Applying all four conditions, we find
A = α + δ,
(65a)
B = δ,
(65b)
Ae−ξexit + Bξexit e−ξexit − δ = 1,
−ξexit
(−A + B − Bξexit )e
Therefore, from (66), one readily finds
ξexit = ln
= −2.
δ .
δ−1
(66a)
(66b)
(67)
In turn, this yields
δ − 1 δ − 1 2δ ln
1+
,
(68)
δ−1
2
δ
for all δ > 1, precisely the threshold value stated
√in formula
(17) in Theorem 3.
1
√
Second, we solve system (63) for all c ∈ 2, δ + δ and δ > 1, with α ∈ (α∗δ , ∞). For c > 2, the
α∗δ =
equilibrium (−δ, 0) is a stable node, with strong stable direction v2 = − d1 u2 and weak stable direction
v2 = −du2 . The general solution is
ξ
u2 (ξ) = Ae−dξ + Be− d − δ.
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Wave speeds for the FKPP equation
627
Also here, we may set ξ = 0 as the time of entry into Region 2, and we again denote the time of exit
from Region 2 by ξexit . Hence, the desired solution satisfies u2 (0) = α, v2 (0) = −dα, u2 (ξexit ) = 1, and
v2 (ξexit ) = −c. Applying all four conditions, we find
δ
,
(70a)
A=α+ √
d c2 − 4
dδ
B = −√
,
(70b)
c2 − 4
ξexit
(71a)
Ae−dξexit + Be− d = 1 + δ,
B ξexit
dAe−dξexit + e− d = c.
(71b)
d
Next, imposing the requirement that d(1 + δ) − c > 0, we solve the latter two equations as a linear system
ξexit
for e−dξexit and e− d :
B 1+δ
−dξexit
e
−c ,
(72a)
=
D
d
ξexit
A
e− d =
(c − d(1 + δ)) ,
(72b)
D
√
where D = AB c2 − 4 < 0. Then, using the formulas for A, B, and D (and recalling that A > 0, B < 0,
and D < 0), we solve these two equations for ξexit and equate the expressions to derive the following
relation:
δ
δ
1
δ
1
−d ln dδ α+ √
(1+δ − cd) − d ln
α+ √
= ln √
(d(1+δ) − c) .
d
d
d c2 − 4
d c2 − 4
c2 − 4
(73)
Finally, simplifying this relation and solving for α as a function of c and δ, we find
(1 + δ − cd)d
δ
√
√
α(c) =
−1
d c2 − 4 (δd)d c2 −4 (d(1 + δ) − c)d2
(74)
This is the desired relation, (18), between α and c, for all δ > 1, stated above in Theorem
3.
√
We observe that the requirement d(1 + δ) − c > 0 is equivalent to requiring c < δ + √1δ . Hence,
√
1
√
c ∈ 2, δ +
.
(75)
δ
Geometrically, the upper boundary in the phase plane corresponds to the case in which solutions on
the weak stable direction, v2 = −du2 exit Region 2 exactly at the point (1, −c), which is on W s (Q+ ).
Hence, this is the largest value of c for which the solutions on the manifold W u (Q− ) can connect to Q+ .
For larger values of c, solutions on the weak stable direction exit Region 2 above that point, and hence,
W u (Q− ) cannot asymptote to Q+ , and no singular heteroclinics of this type are possible.
d(−c)
< 0 and d(−dα)
> 0, so that the points P2∗ and P2in∗
Next, we establish that dα
dc > 0. Clearly,
dc
dc
move in opposite directions (down and up, respectively), as c increases. Let 2 < c1 < c2 . As c increases,
the vector field in (63) rotates clockwise, since the Jacobian of
v2
v2
(76)
−c1 v2 − (u2 + δ) −c2 v2 − (u2 + δ)
2
δ
equals
√ (c1 −c2 )v2 , which is negative. Hence, for any chosen α > α∗ , there exists only one value of c in
1
2, δ + √δ at which the desired singular heteroclinic connection exists. Therefore, applying standard
phase plane analysis, see, for example, Proposition 2 in [17], we find that
dα
dc
> 0.
628
F. Dumortier and T. J. Kaper
ZAMP
Finally, we observe that the proof of the persistence of the singular heteroclinic orbit for 0 < ε 1
and the rigorous asymptotics for c̃ given in Sect. 3 carry over to this system, as well. This completes the
proof of Theorem 3.
Acknowledgments
The authors thank Wim van Saarloos for motivating this work and for helpful conversations. The authors
also thank an anonymous referee for suggesting a number of important improvements in the article,
especially to expand the statements of Theorems 2 and 3. F.D. thanks Boston University for its hospitality
and support during the period in which the research was conducted. The research of T.K. was supported
in part by NSF Grant DMS-1109587.
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Freddy Dumortier
Universiteit Hasselt
Campus Diepenbeek
Agoralaan Gebouw D
3590 Diepenbeek, Belgium
e-mail: [email protected]
Tasso J. Kaper
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215, USA
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(Received: July 22, 2013; revised: February 12, 2014)