TICAM REPORT 97-03
February 1997
Existence and Asymptotic Properties of
Solitary-Wave Solutions of Benjamin-Type
Equations
Hongqiu Chen and Jerry L. Bona
EXISTENCE AND ASYMPTOTIC PROPERITIES OF
SOLITARY-WAVE SOLUTIONS OF BENJAMIN-TYPE EQUATIONS
HONGQIU
CHENI
AND JERRY
L.
BONA 1,2
Abstract. Benjamin recently put forward a model equation for the evolution of waves on the interface
of a two-layer system of fluids in which surface tension effects are not negligible. It is our purpose here
to investigate the solitary-wave solutions of Benjamin's model. For a class of equations that includes
Benjamin's model featuring conflicting contributions to dispersion from dynamic effects on the interface
and surface tension, we establish existence of travelling-wave solutions. Using the recently developed
theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions.
1.
Introduction.
Considered here is the evolution of waves on the interface of an idealized incompressible,
two-fluid system consisting of a light fluid of density PI and depth hI, resting on a heavier
fluid of density P2 and depth h2, bounded above and below by rigid horizontal planes. We
assume that P2 > PI, hI » h2, and focus attention on wave motion that doesn't vary in
the direction perpendicular to the primary direction of propagation.
The fluid domain is
described quantitatively by a standard x-y-z-Cartesian coordinate system so oriented that
gravity acts in the -z-direction,
the interface between the two fluids at rest is located at
z = 0, the primary direction of wave propagation is along the x-axis, and so the dependent
variables describing the fluid motion do not depend on the independent variable y.
If the surface tension is neglected and the waves are uni-directional, of small amplitude
and long wavelength, the fluid motion can be described approximately by the BenjaminOno equation
1]t
+ 1]x + 1]1]x -
L1]x
= 0,
where L = H8x is the composition of the Hilbert transform and the spatial derivative in the
direction of primary propagation, or, equivalently, L is a Fourier multiplier operator with
symbol I~I, which is to say Lv(~) = 1~lv(~).Here, 1] = 1](x, t) is the vertical displacement
of the interface between the two fluids at the spatial point x at time t, and the equation
is written in a non-dimensional, scaled form (see Benjamin 1992, Benjamin 1996, Albert
et at. 1997).
If surface tension cannot
same level of approximation
be safely ignored, the interfacial
by the Benjamin equation,
1]t
+ 1]x + 1]1]x -
aL1]x
± {31]xxx
waves are described at the
= 0,
1Department
of Mathematics, The University of Texas at Austin, Austin, TX 78712.
2Texas Institue for Computational and Applied Mathematics, The University of Texas at Austin,
Austin, TX 78712.
Typeset by A,MS-
'lEX
where L is as above and a and {3 are non-negative constants. For a
the circumstances under which this equation is likely to be physically
et at. (1997, Section 2). Define the new dependent variable u by
1](x, t) = u(x + t, t) if the plus-sign appears and u(x, t) = -1]( -x - t,
holds; in terms of u, Benjamin's equation becomes
Ut + uUx ± aLux
+ {3uxxx
=
o.
detailed analysis of
relevant, see Albert
the transformation
t) if the minus-sign
(1.1)
As written in (1.1), the equation reduces to the Korteweg-de Vries-equation (KdVequation henceforth) when a = a (dispersive effects are dominated by surface tension),
and to the Benjamin-Ono equation (BO-equation henceforth) when {3 = 0 (negligible
surface tension). In this paper we are interested in solitary-wave solutions of (1.1) in case
both a and {3 are non-zero. Unlike the situation that obtains for the KdV-equation, the
solitary-wave solutions of (1.1) feature algebraically- rather than exponentially-decaying
outskirts. Moreover, depending on the sign in front of aLux, the tails of the solitary waves
may feature a finite number of oscillations as they evanesce.
Provided {3 > 0, a rescaling of x and t allows us to assume (3 = 1 in (1.1) without loss of
generality. To study solitary-wave solutions, it is natural to substitute the travelling-wave
form u(x, t) = ¢(x - ct) into (1.1). After a few simple manipulations, this format for u
leads to the operator equation
(1.2)
¢=A¢,
where the constant c
defined by
>a
velocity of the wave and the operator A is
is the propagation
1
00
A¢(x) = -(c
1
2
+ .C)-I¢(X)
=
k(x - y)¢2(y) dy,
(1.3a)
-00
with the kernel k given explicitly in terms of its Fourier symbol
k(~)=
_.~
1...
(1.3b)
Our purpose here is several-fold. In Section 2, we bring to bear techniques from nonlinear functional analysis, notably positive-operator ideas and concentrated-compactness
methods to establish a satisfactory theory of existence for solitary-wave solutions of (1.2).
Section 3 is devoted to a rigorous analysis of the decay of these solitary waves. There are
conflicting formal analyses of this aspect by Benjamin (1992, 1996), and our theory settles
several aspects of the spatial asymptotics of these waves. In Sections 4, 5 and 6, we take
up a natural generalization of equation (1.2) which features at the same time more general
nonlinearity and more general forms of competing dispersion. The theory is broadened
to include existence and decay results for solitary-wave solutions of this general class of
nonlinear, dispersive wave equations.
2
2. Existence of Solitary Waves.
The discussion of existence of solitary waves is broken into two parts. First, consideration is given to the case of a plus sign in front of the term aLux. For this situation,
positive-operator methods come to the fore and existence is seen as a consequence of extant theory. Section 2.2 treats the more interesting case where a minus sign appears, and
the dispersion arising from surface tension competes with the usual frequency dispersion
brought on by finite-wavelength effects. Here, methods of concentrated-compactness
(see
P.-L. Lions 1984) win the day.
2.1. The Symbol of.c
is e + al~1
To establish existence in case the kernel k has Fourier symbol _.. ~
1
. where
a > 0, we rely upon the theory of solutions of nonlinear convolution equations of the
form (1.3) put forward in Benjamin et al. (1990). The conclusion of the theory is that,
provided the kernel k = kc satisfies appropriate hypotheses, equation (1.3) has a solution
¢(x) = ¢c(x) which is an even function, decaying monotonically to zero as x --t ±oo.
Moreover, ¢ and all its derivatives lie in LI (R) n Loo (lR). In fact, more is true as will be
seen presently.
The following technical lemma dmonstrates that the kernel satisfies the hypotheses
needed for the validity of the principle results in the last-quoted reference.
LEMMA
2.1. Let k be defined via its Fourier transform as in (1.3b) for the case of a plus
sign. Then k has the following properties:
(1) k is a real-valued, even, bounded, continuous function and k(x) --t 0 as Ixl
(2) k is positive on IR, k E LI (IR),
(3) k is monotone decreasing on (0, (0), and k is strictly convex for x> O.
--t 00,
Proof. Write k as a Fourier integral thusly:
k(x)
=~
.j2:;
1
00
eiXek(~) d~
=~
-00
roo
.j2:; io
cos(xO
c+
a~ + e
d~ .
From this representation, it is clear that k is real-valued and even. Since k is the Fourier
transform of an £1 (IR)-function, the Riemann-Lebesque lemma assures that k is bounded,
continuous and tends to zero at ±oo. To establish the rest of the properties, it is convenient
to represent k in another way.
Consider the complex function
eiwz
j(z)
= c+az+z'>'
3
where w 2: 0, and z = x
+ iy.
> 0,
Since c, a
n = { z = pew
:
0 ::::; p ::::;R,
r
Jan
an in
Parametrizing
1+
o
c
eiwx
ax
+ x2
dx
+
1~+
c
0
r~
I Jo
yields
()
o.
= O.
eiwRei8
1
.j2:;
.
..
iRe~B de +
aRe~B + R2e2~B
iwRei8
w
c + a~eiB + R2e2iB iRe
1
0
e-WY
1,ay - y?
R c
idy = O.
del-t a as
R
-t
2.1
()
00.
(2.1) and then passing to the limit as R -t
00
0
1 +.
lemma,
Taking the real part of the relation
k w =-
f(z) dz
the obvious way reveals that
As in the usual proof of Jordan's
for w 2:
e ::::;~} .
0::::;
theorm, for any R > 0,
By Cauchy's
R
j is analytic in the closed region
cos( wx)
c+ax+x2
dx=-
1
.j2:;
1
00
0
00
thus
WY
aye(c_y2)2+a2y2
dy
'
Because k is even. it follows that for all x E IR,
(2.2)
Clearly, the representation
it is easily seen that
(2.2) entails that k(x)
>a
for all x E IR. Moreover, for x
>
0,
and that
Thus k is monotone decreasing and strictly convex on (0, (0), and the lemma is proved. 0
4
2.2. For a, (3 > 0, the Benjamin equation with the +-sign in front of the
term aLux has a non-trivial, solitary-wave solution u(x, t) = ¢c(x - ct) for each c > O.
The solitary-wave ¢c may be chosen to be an even, positive function, strictly monotone
decreasing on (0, (0), and such that ¢c and all its derivatives are bounded, continuous
£1 (IR)-functions.
THEOREM
Remark. Theorem 2.2 follows immediately from Benjamin et al. (1990, Theorems
and 3.9 ), because the hypotheses of this result are a consequence of Lemma 2.1.
2.2. The symbol of £
3.7
is e - al~1
To assure the operator A has a kernel k that is free of certain types of singular behavior,
the parameter a must be less than 2y1c. This restriction is already discussed in detail in
Benjamin (1992). Proceeding as in Section 2.1, but using the Residue Theorem instead of
Cauchy's Theorem, the symbol k is determined to be
k x _ __ 1
( )-
f2=
V £.'!r
1
(C - yay
2)2 +
0
f2=
aye -Ixly
00
2 2
dy
.
V£.'!r
+ J 4c -
a
2
e
a
-~Ixl
2
cos -x.
2
Because the kernel k fails to be non-negative, an existence proof using positive operator
theory is not available. We turn instead to the concentration-compactness
theory (see
Lions 1984, Weinstein 1987).
Consider the equation
(2.3)
Following Weinstein,
we introduce
the functional
AU) =
A for our variational
analysis, namely
J f(c + £)j
(f j3(x) dx)~ .
Throughout, an unadorned
functional J on HI (IR) by
integral will always means an integral over R. Define also the
J(u) =
J
u(c
+ £)u,
and for>. > 0, let
8(>')
Remark: The restriction>.
then so is -u.
=
>
inf{ J(u) : u E HI(R),
a
J
u3(x) dx = >.}.
can be replaced by >. #
5
a because
(2.4)
if u is a solution of (2.4)
LEMMA 2.2. (1) The equation (2.3) is solvable if min{AU) : f E HI, f # a} is solvable;
(2) Problem (2.4) is equivalent to min{AU) : f E HI, f # a}; more precisely, if f is a
minimizer
of 8(>') for some>. > 0, then a simple rescaling ¢ = 28,(>') f
= 28(1)-4-
>.3
A
(2.3).
Proof. (1) If f is a minimizer of AU),
For hE Hi, we have
solves
then the Frechet derivative of A at f must vanish.
0= A'(f)h
2
2(J f3(x)dx)3
1
J f(c+£)hdx-2J
f(c+£)fdxJ
(J j3 dx)
whence
J
f(c+£)hdx=
j2hdx(f
f3(x)dx)-3
4
3
J fj;3~~dX
J
2
f hdx
for any h E HI. It follows that
(c
+ £)f
+ £)f dx 2
Jj3dx
f,
= J f(c
is the same as (2.3) except for the coefficient. If we define ¢ by
and this equation
2 J f(c + £)f dx f = 2AU)
f
¢=
J f3 dx
(J f3 dx)
1
= 28(1)
U,
l
>. 3
3
then ¢ does not depend on >. and satisfies equation (2.3). 0
The next two results are taken from Lions (1984) and repeated
convenience.
here for the readers'
LEMMA 2.3. Let {Pn}n~I be a sequence of non-negative functions in LI(RN)
Pn(x) dx = >. for all n and some>. > O. Then there exists a subsequence
satisfying one of the following three conditions:
In~.
(1) (Compactness) there are Yk E IRN for k = 1,2,· .. , such that Pnk (.
which is to say that for any E > 0, there is an R large enough that
(2) (Vanishing)
for any R > 0,
limk-+oo
SUPyEIRN ~x-YI~R
6
Pnk (x) dx = 0;
satisfying
{Pnkh~I
+ Yk)
is tight,
(3) (Dichotomy)
there exists a E (0, >') such that for any
N
and Pk, P~ E LI(R
IIPnk - (Pk
I
Pk, P~ 2:
),
J Pn(x)
>
0, there exists ko 2: 1
such that for k 2: ko,
+ p~) IIL1 :::; E,
r Pkdx-al::::;E,
J~N
I
supp Pk n supp P~ =
Remark:
a
E
0,
r p~dx-(>.-a)I:::;E,
J~N
dist{ supp Pk, supp pn --t
J Pn(x)
In Lemma 2.3 above, the condition
dx = >'n where >'n --t
>. >
00
as k --t
00.
dx = >. can be replaced
by
O. To see this simply replace Pn by ~: and apply the
lemma.
2.4. Let 1 < p ::::;
00 and 1 ::::;
q < 00. Assume for n = 1,2,···
in Lq(IR), u~ is bounded in Lp(IR), and for some R > 0,
LEMMA
as n --t
00.
Then it transpires that for any r
Un
--t
, that Un is bounded
> q,
0 as n --t
00
in Lr (IR).
2.5. Let c > a be a given wave-speed such that minxE~{ x2 - alxl +c} > 0, and
let>. be any positive number. Then every minimizing sequence {Un}n;:::1of the problem
(2.4) is, up to a translation in the underlying spatial domain, relatively compact in HI (IR).
Hence, there is a solution of the problem (2.4), and therefore there exists a non-trivial
solution of problem (2.3) by Lemma 2.2. Thus the Benjamin equation has solitary-wave
solutions corresponding to the wave speed c.
THEOREM
Proof. We begin with a couple of observations
follows that
and, in any event,
J(U)
J
::::; (k2 +
about the functional
c)lill2
7
:::;
tllullk1'
J. Since a < 2y1c, it
where '1 = ~(1- ~~)min{l,c},1'
the minimizing sequence {un }n~
= max{l,c}.
This means that 0
is bounded in HI (IR).
1
<
8(>.)
<
CXJ
and that
Denote by Pn the quantity lunl2 + lu~12,and let Pn(x) dx = µn· Then µn is bounded
and furthermore, µn = Ilunll~1 2: IluliL 2: (fu3(x)dx)~
= >.~, because HI C L3 with an
embedding constant less than one.
J
Without loss of generality, suppose µn --t µ as n --t 00. To prove the theorem, we
apply Lemma 2.3 to the sequence {Pn}n~I, after ruling out the possibilities of Vanishing
and Dichotomy. Suppose there is a subsequence {Pnk h~Iof {Pn}n~I which satisfies either
Vanishing or Dichotomy. If Vanishing occurs, which is to say for any R> 0,
lim sup
k-+oo yEIR
1
Pnk (x) dx = 0,
Ix-yl~R
then
so, by Lemma 2.4,
This leads to a contradiction
since
If Dichotomy occurs, there is aPE
ko and Pk, P~ E LI (IR), Pk, P~ 2:
Ilpnk - (pl
I
L pl
supp
dx -
a
+ p~) IIL1:::;E,
pi ::::;E,
pl n supp
I
(0, µ) such that for any
E
> 0 there
such that for k 2: ko,
L
P~ dx - (µ - µ) I ::::;E,
(2.5)
P~ = (/)and dist{ supp pl, supp pn --t
00
Without loss of generality, it may be supposed that the supports of
as follows:
supp
pl
C (Yk - Ro, Yk
C
IR and Rk --t
To obtain splitting functions uk and u~ of unk'
:::;(, 'ljJ :::; 1 be such that
((x) = 1 when
'ljJ(x)
=
a
as k --t
pl
00.
and P~ are separated
+ Ro),
for some fixed Ro > 0, a sequence {ydk~I
Lions 1984).
o
corresponds a
Ixl :::;1,
00
k = 1,2,···
(see the construction
, let (, 'ljJ E
((x) = a when \xl 2: 2,
when Ixl :::;1,
'ljJ(x) = 1 when
8
Ixl 2: 2.
ego
in
with
=
Denote by (k(X)
enough that
((X~~k),
=
'l/Jk(X)
'l/J(X~~k),
for x E R, where RI
>
Ro is chosen large
(2.6a)
and
I
J
2
+ lax('l/Jkunk)j2
l'l/JkUnkl
- p~) dxl ::;
(2.6b)
£.
To see this is possible, first note that from (2.5),
£
2: Ilpnk - (pl + p~)
r
=
IILI
pll
Ipnk -
+
dx
J1X-Ykl'5:.Ro
r
IPnk - p~1 dx
+
J1X-Ykl?2Rk
r
Pnk dx,
JRo'5:.lx-Ykl'5:.2Rk
whence
1
IX-Ykl'5:.Ro
IPnk - pll dx ::::;£,
r
Pnk dx ::::;£.
J Ro'5:. IX-Yk 1'5:.2Rk
In consequence of these relations, it transpires that
J
I
'I
=
=
12 + lax ((kunk
(I(kunk
IX-Yk 1'5:.2Rl
r
J'X-Ykl'5:.Ro
+ r
J Ro'5:.lx-Yk
I
::::;r
J1X-Yk I'5:.Ro
2
(l(kUnk
1
2
(Iunk
1
1'5:.2Rl
)12
+ lax
+ IU~k
(l(kUnk
Ipnk - pll dx
pl)
dxl
((kUnk
)1
-
2
1
-
2
-
pl) dxl
pl) dXI
12 + 1(~12Iunk 12
+ max{l(k(x)12
+ l(kI2Iu~k
+ 1(~(x)12}
xEIR
12) dx
r
J Ro'5:. IX-Yk 1'5:.2Rl
Pnk dx
::::;£ + £ = 0(£)
as
£ --t
I
J
=
0, and
(I'l/Jkunk
2
1
11
+ lax('l/Jkunk)12
(l'l/JkUnk
IX-Ykl?Rk
::::; I
r
J Rk'5:.lx-Yk
2
::; max{I'l/Jk(x)1
xEIR
=
1'5:.2Rk
2
1
- p~) dXI
+ lax('l/Jkunk) 2
1
p~) dxl
-
2 + lax ('l/Jkunk )12 - p~) dxl
(l'l/JkUnk 1
+ l'l/JUx)12}
1
Rk'5:.lx-Ykl'5:.2Rk
0(£)
9
Pnk dx
+
+ 11
1
IX-Yk 1?2Rk
IX-Ykl?2Rk
2 + IU~k 12 - p~) dxl
(Iunk 1
Ipnk - p~1 dx
as E --t O. Thus if we set uk = (kunk' u~ = 'ljJkunk and define Wk by unk = uk + u~ + Wk,
then uk, u~, Wk E HI, J lul(x)13 dx is bounded, and there exists a subsequence of {ukh~I'
still denoted {ukh~I' for which there is a ko > 0 and a, {3 E IR such that for k 2: ko,
IIwkllHI =
=
=
11(1 -
C((, 'IjJ) (
A simple calculation
E --t
+ 1(1- (k(X)
--t 00
as k
-'ljJk(x))'I}
is a constant
--t 00.
+ u~ + Wk)
J(Uk) + J(u~) + J(Wk) - 2a
+2
as
dx) ~
shows that
=
J
Pnk
1:::;2Rk
(k(X) -'ljJk(X)1
ut, supp un
J(Unk) = J(Uk
where Hu' =
r
J RI :::;IX-Yk
O(E)
as E --t 0, where C((, 'IjJ) = maxxElldl1only dependent on ( and 'IjJ, and
dist{ supp
IIHI
(k - 'ljJk)unk
f
uk(c + £)Wk dx + 2
f
f
ukH(u~)'
dx
u~(c + £)Wk dx,
u'(y) dy is the non-local operator whose Fourier symbol is
x-y
0,
J(Wk) ~
l+c
~llwkllHI
2
=
O(E),
If
Uk(C
+ £)Wk
dxl ~
J
u~(c
+ £)Wk
dxl ::::;I'llu~IIHlllwkllHI = O(E),
and
I
I'IIUkllHIllwkllHI = O(E)
10
I~I.Notice
that,
and that
as k --t
00,
because Rk --t
00
and RI is fixed. In consequence,
it is seen that
8(>') = lim inf{ J( un)} = lim inf{ J( unk)}
n
k
= liminf
k
- 2a
f
{J(ul)
J ul(x)3
+2
ulHuk
2: li~inf J(uk)
as c --t O. If
+ J(uk) + J(Wk)
f
uk(c
+ £)Wk + 2
f
+ limkinf J(uk) + O(c)
dx --t (3 = 0, then by (2.6a),
as c --t 0, and therefore
8 (>') > ')'µ + lim inf J (uk)
-
as c --t O. Letting
E
k
--t 0 in the last relation
8(>') 2:
If, on the other hand,
J uk(x)3
a
leads to
1.Jl + 8(>') > 8(>').
dx --t (3 =1= 0, then
8(>') 2: 8({3)
and letting c --t
+ 0 (E)
+ 8(>'
- (3)
+ O( E),
gives
8(>') 2: 8({3)
+ 8(>.
11
- (3).
uk(c
+ £)Wk}
But, for () E IR, 8(()>') =
I()I~ 8(>').
If we write {3 = ()>., then
+ 8((1 - ())>')
= 1()138(>') + 11- ()138(>')
= {1()13 + 11- BI3}8(>')
0> 8(>') 2: 8(()>')
2
2
2
2
> 8(>.),
another contradiction.
Thus Dichotomy is seen to be impossible.
Since Vanishing and Dichotomy have been ruled out, it is concluded that there is a
sequence {Yn}n2:1 C IRsuch that for any E > 0, there are R < 00 and no > 0 such that for
n > no,
1
Pn(X) dx 2: µ -
IX-Ynl~R
11
IX-Yn
1
E,
1
un(x)3 dxl ::::;
I2:R
Pn(X) dx ::::;E,
IX-Ynl2:R
lun(x)13 dx
I>R
IX-Yn
1
::::;Ilunl\Hl
IX-Yn
Pn(X) dx
I2:R
::::;O(E)
as
E --t
O. It follows that
11
un(x)3 dx - >.1 ::::; E.
IX-Ynl~R
Letting un(x) = un(x - Yn) for x E R, the above property means that Un (or a
subsequence) converges weakly in HI, a.e. on IR, and strongly in L3(R) to some HI_
function U, say, and
Furthermore,
8(>') = li~inf
2:
f
u(c
f
un(c
+ £)u
+ £)un
dx
dx.
u
Thus the function
solves the variational problem (2.4) and therefore ¢ =
the problem (2.3). Theorem 2.5 is proved. 0
3. Asymptotic
282')
u solves
decay of solitary-wave solutions.
In the previous section, we discussed existence of solitary-wave solutions of the Benjamin equation. In this section, attention is turned to the asymptotic properties of such
solutions. According to a recent result reported in Bona and Li (1997) they resemble those
of the kernel k. Here is the relevant theorem.
12
LEMMA 3.1. Suppose that
f
E
Loo with limlxl~oo f(x)
equation
f(x)
=
f
=
k(x - y)G(f(y))
a is a solution
of the convolution
dy,
where the kernel k is a measurable function satisfying k E HS for some s > ~ and G is a
function such that IG(x)1 ::::;Clxlr for all x E R, and for some constants C > 0 and r > 1.
Then fELl
n L2 and there is a constant 1 with a < 1 < s such that Ixll f(x) E LI n Loo.
Furthermore,
(1) if there is a constant m > 1 such that limx~±oo Ixlmk(x) = C± for some constants
C± E <C corresponding to limits at +00 and -00, respectively, then
lim Ix 1m f(x)
x~±oo
f
= C±
G(f(t))
dt,
(2) and iflimx~±oo eO'olx1k(x) = C±, then sup eO'olx1If(x)1
<
and
00
for some constants C± corresponding to limits at +00 and -00, respectively.
In the present context, based on Lemma 3.1, the following may be concluded.
THEOREM 3.2. For the problem (1.2) where a
> 0 is
as restricted previously,
for some constant C E R, C # o.
Proof. Obviously,
k
E
HS for any s < ~. Hence it will suffice to find a suitable expression
for k. For the case k(~) =
~f
•
~""
~
k(x) = :F-Ik(x)
....
'>\'
1
it was found earlier that
00
= .j2:;
The change of variables ~ = xy transforms
1
0
aye-1x1y
(c _ y2)2
the right-hand
and thus
13
+ a2y'>
side to
dy.
by the Dominated Convergence Theorem.
For the case k(~) =
The same transformation
,..,1
~"I
...
'>\'
it was seen previously that
~ = xy gives
Applying the Dominated Convergence Theorem again leads to the conclusion
By Lemma 3.1, the theorem is proved. 0
Remark: Notice that this rigorous result shows clearly that Benjamin (1992) is incorrect
in his assertion that the tails of solitary-wave solutions ¢(x) of his model equation oscillate
infinitely often as x --t ±oo. Indeed, a slightly refined analysis shows that these solutions
approach zero monotonically as x --t ±oo, though they do feature a finite number of
oscillations as one sees from the numerical approximations reported in Albert et al. (1997).
4. More General Nonlinearity and Dispersion.
In this section and the next, attention is turned to an extension of the theory developed
in Sections 2 and 3. Consideration is given to the situation where the effects of dispersion
are modelled by a competing pair of homogeneous terms and the nonlinearity is a pure
power. Thus the operator £ is defined by
(4.1)
and the nonlinearity has the form
F(¢) = ~¢P-l,
p-1
where m and p are positive integers, p
> 2,
(4.2)
and r is a real number with 0 ::; r < m.
We aim to establish existence and asymptotic decay rates for solitary-wave solutions
of the nonlinear, dispersive wave equation
(4.3)
14
where F and £ are defined in (4.1)-(4.2). Some of the development can be abbreviated
because it parallels the discussion of Benjamin's equation in Sections 2 and 3.
As before, the assumption that u(x, t) = ¢(x - ct) is a solitary-wave solution of (4.3)
implies that ¢ is a solution of
(c + £)¢ =
1
_rJ>P-I.
(4.4)
p-1
The proof of the following theorem is the goal of the present section. In Section 5, the
issue of the large-space asymptotics of solutions of (4.4) is taken up.
THOREM
4.1. Let c
>
0 be a given wave-speed
and suppose m,p
E
Z+, p
>
2, and
m-r
a ~
r < m. If a lies in the range a
c----m-
<
equation
(4.4) has a solution
¢ E HOO.
Remark: Notice that if ¢ E HI (IR) satisfies (4.4), then necessarily ¢ E HOO (IR). This
follows because the assumption on the range of a implies that the symbol of c + £ has the
property
for all ~. In consequence, there are positive constants '1 and
1such
that
(4.5)
for all ~. Hence c + £ is an isomorphism of Hs+2m(IR)
onto HS(IR) for any s 2: 0, say.
I
Therefore, if ¢ solves (4.4) and ¢ E HI (IR), then <J>P- E HI (IR) and hence
It then follows that
rJ>P-I E H1+2m(IR),
whence
Continuing this argument inductively demonstrates that ¢ E Hoo(IR).
An application of
the theory of Li and Bona (1996) shows also that these solitary waves are the restriction
to the real axis of a function holomorphic in a strip {z: 1~(z)1 < O"} for some 0" > O.
To prove Theorem 4.1, introduce the functional
AU) =
J f(c + £)f
(f fP dx)% '
15
(4.6)
in Section 2, define J for u E Hm (JR.) by
and, in direct analogy with the developments
J(u) =
and 8 for
J
+ £)udx
u(c
>. > 0 by
8(>.) = inf{ J(u) : u E Hm(IR),
J
(4.7)
uP(x) dx = >.}
Remark: Actually, 8(>') = 8(->') if p is an odd number, and so we need only require
>. # 0 instead of >. 2: o. Also, p need not be an integer in our theory. The development
is unchanged if p = r;: where m and n are relatively prime and n is odd, if we choose the
branch of z --t z!i that is real-valued
By the same argument
LEMMA 4.2.
as that in Lemma 2.2, we have the following result.
(1) The problem
(2) minimization
on the real axis.
(4.4) is solvable ifmin{ AU) : f E Hm,
of AU) in (4.6) is equivalent
f #
o} is solvable;
to (4.7), namely, any solution of (4.7) is a
1
minimizer
of (4.6), and if
f
is a minimizer
of (4.6) then the rescaling
a solution of (4.4). Therfore if the problem (4.7) has a non-trivial
solved by the function ¢ defined by the rescaling
which is independent
of
The following representations
>.V
(f fp dx)v
u,
solution
1
f
is
then (4.4) is
any minimizing sequence {Un}n~I of the
in the underlying spatial domain, relatively
on a, (4.5) applies and it is concluded that
in Hm(IR) and that 0 < 8(>.) < 00 if >. > O.
will be useful in the arguments
SUBLEMMA. Let M be the operator defined by Mu(~) =
o < r ::::;m, m a positive integer. It follows that
n E Z+, then
Mu(x)
= (_l)n
f!.J
V -;
16
a;n+lu(y)
x - y
presented
1~lru(~)for
(1) ifr = 2n for n E Z+, then
+ 1 for
H
>..
Proof of Theorem 4.1. It is first shown that
variational problem (4.7) is, up to translations
compact in Hm(JR.). Because of the restriction
any minimizing sequence {Un}n~I is bounded
(2) ifr = 2n
f
d .
y,
presently.
u E Hm,
where
(3) if r = 2n
Mu(x)
+8
for n E Z+, 0
<8<
VF(
2.
81f
= (_l)n
1, then
cos( 2" )f(l - 8)
)
-1
J
sign(x - y)
a~2n+1u(y)
IA
dy,
where f connotes the usual gamma-function;
(4) and ifr = 2n + 1 + 8 for n E Z+, 0 < 8 < 1, then
Mu(x)
=
(-It-I
Proof. Part (1) is obvious.
right-hand
8
)
sin( ~ )f(l- 8)
2
~
~( 2
-1
J
a2n+2u( )
y
Y dy.
Ix -
yl8
Part (2) is proved by taking the Fo~rier Transform of the
side of the expected equality and using the formula
(_l)n
=( -It
I!
I!
F-I{ a;n+1u(x)}
1
- (i~)2n+1u(~) rrc
1f
v21f
=.!.ien+1U(~)(
F-I{
J
J smx x dx
= 1f to obtain
1}
ixe
-ex
dx
- isign(~)1f)
1f
=1~12n+lu(~).
(3) When r = 2n
relation is
+ 8,
the Fourier Transform of the right-hand
side of the desired
roo sin~x
8~)
.
.
where the formula Jo -y- dx = cos( - )f(l - 8 can be found m Oberhettmger
x
2
(1957, p.5).
(4) When r = 2n + 1 + 8, then the Fourier Transform of the right-hand side of the
17
advertised equality in (4) is
roo cos~x
---r-
where the elementary formula Jo
Oberhettinger
The sub lemma is established.
Attention
the quantities
x
(1957, p.116).
.
8~
2
.
dx = sm( - )r(l - 8) can be found m
0
is now given to finishing the proof of Lemma 4.2. Denote by Pn and µn
Pn(x)
= lun(x)12 +
la;mun(x)12
and µn
=
J Pn(x)
.£
dx.
Then {µn}n~I
is
2
bounded, and µn = IIPnllLl 2: IlullEp 2: (J un(x)P dx) p = Xi>, because Hm(IR) c Lp(R)
with an embedding constant less than one for any p 2: 2. Without loss of generality, suppose
µn --t µ as n --t 00. To prove the theorem, apply the concentration-compactness principle
Lemma 2.3 to the sequence {Pn}n~I and aim to rule out the possibilities ofVanishiing and
Dichotomy. Suppose there is a subsequence {Pnkh~I
of {Pn}n~I which is either Vanishing
or Dichotomous. If Vanishing occurs, which is to say for any R> 0,
lim sup
1
Pnk (x) dx = 0,
k-+oo yEIR Ix-yl~R
then,
On the other hand, we already know that {u~ k
2.4,
as n --t
00.
This leads to a contradiction
since
Thus Vanishing does not occur.
18
h>I
_
is bounded in L2(IR),
so by Lemma
If Dichotomy occurs, then for any
such that, for k 2: ko,
Ilpnk - (Pk
If
+ p~) IILI ~
Pk(X) dx -
supp Pk
> 0, there
is a ko
> 0 and pl, P~
E
LI(R),
Pk, P~ 2: 0
E,
pi ::::;
E,
n supp
E
If
P~ - (µ -
µ)I ::::;
(4.9)
E,
P~ = 0 and dist{ supp Pk, supp pn
--t
00
as k --t
00.
As in the proof of Theorem 2.5, the supports of Pk and P~ may be taken so that
supp Pk C (Yk - Ro, Yk
+ Ro),
+ 2Rk,
supp P~ C (-00, Yk - 2Rk) U (Yk
for some fixed Ro > 0 and sequences {ydk~I,
{Rkh~I
C JR., where
Rk --t
(0)
00.
To construct the splitting functions uk and u~ of unk for k = 1,2, ... , let (, 'l/J E Cgo
be as defined in Section 2.2, and (k(X) = ((X"R.~k),'l/Jk(X) = 'l/J(X"R.;k) for RI > Ro chosen
large enough that
(4.10a)
and
(4.10b)
The reason (4.10a) and (4.1ob) obtain for large RI is the same as argued earlier in Section
2.2.
u~ = 'l/Jkunk' Wk = unk - Uk - u~ or unk = Uk + u~ + Wk. Then
ul, u~, Wk E HI and the supports of uk and u~ lie in (Yk - 2Rb Yk + 2RI) and (-00, Yk 2RI) U (Yk + 2Rk, (0), respectively. Moreover, J uk(x)P dx is bounded, so there is a subsequence of {ukh~b
still denoted by {ukh~I' converging to some number {3, say. Then for
Let uk
any
E
>0
= (kunk'
and k sufficiently large, we have
and
IIWkllkm
=
11(1-
=
fl
(k - 'l/Jk)Unk Ilkm
2
(1 - (k - 'l/Jk)Unk 1 + lam ((1 - (k - 'l/Jk)Unk)
rJ IX-Yk I~2Rk (Iunk 12 + lamunk
= C((, 'l/J) r
Pnk dx
JRl~lx-Ykl~2Rk
~ C((, 'l/J)
Rl ~
::::;O(E)
19
12),
dx
2
1
dx
as E --t 0, by the first inequality in (4.9), where C((,'Ij;) is a constant only dependent on
(, 'Ij;. As before
+ J(u~) + J(Wk) -
J(u) = J(ul)
+2
f
uk(c
+ £)wn + 2
f
2a
f 1~12r;;r(~)~(~)
u~(c
d~
+ £)wn.
Here, there appears the non-local operator defined by the Fourier symbol
integer. In any event, we always have
If
J(Wk)
::::;'rllwkll~m
Uk(C + £)wkl dx ::;
=
1~12rif r
is not
O(E),
f (c - al~12r + em)I;;r(~)IIWk(~)1
d~
::::;'rIIUkIIHm IIWkllHm
=
and
If
u%(c
+ £)wkl
O(E)
f (c - al~12r + em)I~(~)IIWk(~)1
dx ::::;
d~
::::;'rllu%IIHm IIWkIIHm.
=
as
E
--t O. To deal with integral
O(E)
-- --
J 1~12mul(~)u~(~) d~, use is made
(1) If 2r = 2n for some n E Z+,
therefore
e
r
of the Sublemma.
is the symbol of a differential operator,
due to supp uk n supp u% = 0.
(2) If 2r = 2n + 1 for some n E Z +, then m 2: n
and hence
20
+ 1,
and
since Rk -t 00 with RI > 0 fixed.
(3) If 2r = 2n + 8 for n E Z +, 0 < 8 < 1, then
f
d~=
1~12r~(~)~(~)
f
(-It
=
(-It
=
C(8)
f R(~)R(~)1~18
d~
f
1~12r~(~)~(~)
f
a~ul(x)
Jo
an+lu2 (y)
sign(x - y) ~
oo
where C(8) = (-1)n(2coS(8;)
If
d~1::::;C(8)ll
1~18d~
((i~t~(~)) ((i~)n~(~))
y-8e-Ydy)-\
IX-Yk 1~2Rl
a~Uk(X)
{Ill
::::;C(8)lIunkIIHr
IX-Ykl~2Rl
kiF.
dydx
so
1
I a;u~(I;l8
IX-Yk 1~2Rk X - Y
IX-Ykl~2Rk
X
_
a < 8 < 1, then
Thus in all cases, it is observed that
8(>') = lim{ J(un)}
n
= limkinf{
- 2a
J
J(uk)
ukMu~
2: liminf J(ul)
k
= lim{
k
J(unk)}
+ J(u~) + J(wn)
+2
J
uk(c
+ £)Wk + 2
+ liminf J(u~) + O(E)
k
21
J
u%(c
1
}2
112+28 dXdy
Y
-to,
as Rk -t 00.
(4) 2r = 2n + 1 + 8 for n E Z+,
dy dxl
+ £)Wk}
as
E
--t O. If
J ul(x)p
dx --t (3 = 0, then
liminf J(ul)
k
>
liminf111ulllHffi
k
= liminf111plliLl
k
> 0,
so
8(>') 2: "Iii + lim inf J( u~)
-
as
E
k
+ O( E) 2: -"Iii + 8(>.) + O( E)
--t 0, and therefore
8(>')
for sufficiently small values of
E.
If
J ul(x)P
8(>') 2: 8({3)
and letting
E
>
8(>')
dx --t (3 # 0, then
+ 8(>. - (3) + O(E),
--t 0 gives
8(>.) 2: 8({3)
+ 8(>'
- (3).
2
As before, 8(e>.) = ev 8(>.) for e > O. Writing (3 = e>., we have
o > 8(>.)
+ 8((1 - e)>.)
2
2
= ev 8(>') + (1 - B) v 8(>.)
= {ev + (1- e)v}8(>'
2: 8(e>.)
2
2
> 8(>.),
another
contradiction.
Thus Dichotomy
is seen to be impossible.
Since Vanishing and Dichotomy have been ruled out, it is concluded that there is a
sequence {Yn}n2::1 C IR such that for any E > 0, there is an R < 00 satisfying
1
Pn (x)
dx > µ -
E,
IX-Ynl~R
or, what is the same,
1
Pn(X) dx <
E,
IX-Ynl2::R
for n sufficiently large. Reinterpreting
in terms of
whence
22
Un,
this amounts to
Denote by Un the translated function unO = Un(' - Yn)' The above estimates mean
that the u~ (or a subsequence) converges weakly in Hm and strongly in Lp to some function
U E Hm, and
f
u(x)P dx = lim
n-+oo
Furthermore,
8 (>') = lim inf
n-+oo
2:
Thus the limiting
(p-~e('\)
function
f
u(c
f
f
un(x)P dx = >..
Un ( c
+ £)u
+ £) Un dx
dx.
U solves the variational
problem
¢ =
(4.3), and therefore
1
1'-2
U solves the problem
(4.1).
Lemma 4.2 is proved and with it Theorem
4.1. 0
4.3. Problem (4.1) has a solution in Hoo(IR) when £ is defined by the Fourier
symbol al~12r + em with m a positive integer, 0 ::::;r < m and any a 2: o.
THEOREM
The same proof is as that put forward in Lemma 4.2 concludes Theorem 4.3.
Remark: As mentioned before, the nonlinear term ¢P may have p non-integer provided
p =
m, n relatively prime and n odd so that yp E IR if y E IR. For the convolution
equation (4.4), the theory would still be available if the nonlinear term had the form
¢ql¢IO' where q 2: 0 is an integer, u > 0 is real, and q + u 2: 2. In Section 6, we offer brief
commentary on a more general class of dispersion operators than those considered thus
far.
r;:,
5. Asymptotic Decay of Solitary-Wave Solutions.
To determine the asymptotic property
tion 4, write problem (4.2) in the form
of the solitary-wave
1 (c + £)-I¢P-I = - 1
¢ = -1
p-
p-1
f
k(x - y)q?-I(y)
as in (4.4). The kernel k is the inverse Fourier transform
of k(~) =
m > 0 is an integer and r is a real number in the range 0
minxEIR{c ± alxl2r + Ixl2m} > O.
THEOREM
solutions discussed in Sec-
<
r
dy,
c±a~
<m
I
(5.1)
112
r+l"
with a 2:
c2'
m
where
a such
that
5.1. Suppose ¢ is a solitary-wave solution of (5.1).
(i) If r is a positive integer or if a = 0, then there is a Uo >
23
a such
that for any u < uo,
as x
--t
±oo.
(ii) Otherwise, there is a constant µ such that
as x
--t
±oo.
Proof. To prove the theorem, it is sufficient to determine the spatial asymptotics of the
kernel k in (5.1) and then appply Lemma 3.1. To this end, it is useful to write k in a more
eizw
suitable form. Let fw(z) = 2
2
' and without loss of generality, suppose w 2: o.
z m - az r + c
The proof breaks natually into three parts.
(1) If 2r is an even positive integer or a = 0, then fw is analytic in the entire z plane
except for 2m poles. Since a and c are real, the 2m poles divide evenly between
the upper- and lower-half plane. Thus we may order the poles {Zj }J~\ so that
Imzj = Yj > 0 for j = 1,··· , m. The Residue Theorem leads to
(5.2)
for w
> O. Since
k is an even function, it follows readily from (5.4) that
lim e17lx1k(x)
x-t±oo
for any a < ao = minI~j~m {Yj} > O.
(2) If 2r is an odd positive number and a
n=
{z :
Z
=
~
pe~
:
0::::;
=0
# 0, let n be
p::::; R,
the closed quarter-disc
7r
0::::; () ::::; 2"}'
Then the function f w is analytic in n with finitely many simple poles, {Zj =
Xj + iYj : j = 1,2,··· , k}, say. Appealing again to the Residue Theorem, it is
determined that
(5.3)
24
Since k is an even function,
k(x)=
~l
_
7r
(1)~
-
OO
0
(5.3) implies
2r
-xy
Y e
2
(c + (_1)my2m)2
dy+2J2;~{i"C'e-YjIXI+iXjIXI}
+ a2y4r
~
J
(5.4)
J
for all x E R. Changing variables in (5.4) gives the representation
roo
2
(-1) 2r:p y2r e-Y
J0
k (x) = I __I ?..-L 1
+ Re{
47ri
(c
+ (-1)m ( ~ ) 2m ) 2 + a2 ( ~ ) 4r
L Cje-YjW+iXjW
dy
}
j
of k. Since the yj are all positive, it is straightforward
~ 21
to determine
that
00
=
IxI2r+lk(x)
lim
x-t±oo
(-1)
2
Y 2 r e-Y dy.
-
c2
0
(3) Lastly, if 2r > a is is not an integer,' the origin of the x - y plane is a branch
point. Cut the plane through the negative x-axis, and consider the branch of the
logarithm which makes 12r = 1. Define the domain
n=
{z
=
'8
pe~
:
7r
E::::; p::::; R, 0::::; () ::::; 2'},
where E > 0 is sufficiently small and R > 0 is sufficiently large that there is no
number outside n and inside the first quadrant which makes z2m - az2r + c equal
to O. As before, an application of the Residue Theorem leads to
1
00
k(w)
=
1<>=
2
V
27r
V
~Re
27r
=
{
roo fw(iy)i
J0
1
{o
2
.j2:;
~1
+ L
-7r
dx
00
= -Re
=
fw(x)
0
(iy)2m
00
0
Re
+
dY}
~i
27r
V
ieiw(iy)
- a(iy)2r
+c
L' resi{f(z)
: Zj}
J
dy
+
~
,
C·e~XjW-YjW
J
-a sin( r7r)y2r e-WY
(c - a cos(r7r)y2r
+ (_1)my2m)2
(5.5)
J
+ a2
sin(r7r)y4r
dy
CjeiXjW-YjW,
j
where Cj are some complex constants dependent on the singular points of fw(z).
Using the fact that k is even, and with the change of variables y I-t Jxly, we obtain
(5.6)
25
It is thus clear that
·
I1m
x~±oo
1
00
Ix 12r+lk( x ) = -
2asin(r7r)
2
c
y 2re -y d y.
0
as x --t ±oo. Applying Lemma 4.1, Theorem 5.1 is proved.
o
6. Further Discussion.
In the previous sections, we discussed the generalized version of Benjamin's equation
where the nonlinear term is a pure power and the effects of dispersion are modelled by homogeneous terms in which the highest-order term corresponds to is a differential operator.
In this section, interest is turned to the situation where
k
£U(~) =
L ajl~12rju(~)
(6.1)
j==1
with the nonlinear term
1
F(¢) = -lfP-I,
p-
(6.2)
as before, where p > 2 is a positive integer and the parameters aj are real numbers with
ak > 0, 0 < rl < r2 < ... < rk, but rk is not an integer. This corresponds to the situation
where the highest-order term in the dispersion relation is not local.
The goal is, as before, to establish existence and asymptotic decay rates for solitarywave solutions of the nonlinear, dispersive wave equation
(6.3)
This amounts to finding a suitably behaved solution of the convolution equation (5.1) where
F and £ are as above and c > 0 is the wave velocity. As the outline of the theory in this
more general case parallels that developed already for the simpler situations considered
earlier, the exposition in this section will be abbreviated, and concentrated on the points
where additional argument is needed to bring the theory to completion.
As before, for
f
E
Hrk (IR) define
A(f) =
J f(x)(c + £)f(~)
dx,
(J f(x)Pdx)p
J(f) =
f
f(x)(c
+ £)f(x)
and, for>. > 0, set
8(>')
= min{J(u): u
E HID,
J
dx,
uP(x)dx
= >.}.
Then as in Lemmas 2.4 and 4.2, a solution of (6.4) yields a solitary wave.
26
(6.4)
6.1. If u is a minimizer of the problem (6.4), then ¢ = ((p -
LEMMA
((p - 1)8(1))
-L
p-2
1
>,-pu
P~2
u =
solves (5.1).
6.2. For any wave-speed c
THEOREM
:;8(>.))
>
0 and dispersive parameters aj, Tj,j = 1,···
,k
satisfying
(1)
2: ~ - ~ and
rk
(2) minx;:::o{c
+ E ajx2rj}
> 0,
every minimizing sequence {Un}n;:::1 of the variational problem (6.4) is, up to a translation
in the underlying spatial domain, relatively compact in Hrk (IR). Therefore the problem
(5.1) has a non-trivial solitary-wave solution ¢ = ¢c E Hrk (R) and furthermore ¢ E
Hoo (IR). Moreover, if
ro = min{rj : rj ~ Z,j = 1,··· , k},
then ¢c satifies
for some non-zero constant C.
Remark:
In Theorem 6.2 above, if Tj E Z for j = 1,2,· .. , then ¢c(x) decays exponentially, which is to say, there exists a (To > 0 such that for all (T < (To, ¢c(x) satisfies
lim
eu1xl¢(xo)
= O.
x~±oo
The proof of this remark is as same as that of the first part of Theorem 5.1.
As in Theorems
2.5 and 4.2, there are two positive numbers '1 and
'1llull~rk::::;J(u)
This means 0
Hrk
<
8(>')
<
00
1such
that
::::;1\lull~rk'
and any minimizing
{Un}n;:::1
sequence
is thus bounded
in
(IR).
To deal with the problem (6.4) via the concentration-compactness
principle in the
present, more general circumstances, it is useful for s > 0 and s ~ Z to endow HB with a
slightly unusual version of its norm. If s = m + 8, where m is non-negative integer and
0< 8 < 1, then
lIull~. =
r..J
r..J
r..J
where
r..J
f
(1 + 1~12)m+6Iu(e)12
d~
J (1 + 1~12m + 1~12m+26)
lIull~m + f 1~1261~12
1f
+
lu(~)
2 d~
1
d~
II
u
112
m
lamu(x) - a u(y)\2
Ix _ yl1+26
Hm
stands for equivalence of norms.
27
d d
Y x,
6.3. Let p > 2 and f E HS for s 2: ~ - ~ = P~2 be non-zero. Then for any R > 0,
there are jo E Z and 1] > 0, only dependent on Ilflltp and R, such that
LEMMA
(jo+~
)R
j (jo-~)R
If(x)IP dx 2:
1].
Proof. It suffices to establish the result in the case s < 1. Using the equivalent norm above,
we have
f
If( x )12d x + JrJr
If(x) - f(y)12 d d Ix _ yl1+2s
Y X -
j(j+~)R
2:
,1
f
+
(If(x)!2
f( )12
1+;S dY) dx =
If()
: _I yl
()-2)R
j=-oo
Ilfllk8
Ilflltp
-
H8 -
f
If( ))P d
x
x,
> 0,
or, what is the same, for any R
00
IIfl12
IIfl12
it 2:
IlfllLP j=-oo
00
j(j+~)R
,1
If(x)IP dx.
()-2)R
Comparing both sides of the last equality, it is seen that there must be a jo E Z for which
(jO+~)R
j (jo-~)R
f
(If(x)12+
If()
f() 12
IIfl12
x - 1+;S dy)dx::::;:;8
Ix - yl
IlfllLP
j(jO+~)R
(jo-~)R
If(x)IPdx,
and hence,
(jO+~)R
j (jo-~)R
If(x) [2 dx+
j(jO+~)R
j(jO+~)R
(jo-~)R
(jo-~)R
I.e.
Ilfll~8 ('
)0
_l)R (' +l)R)
2
,)0
2
Jf(x) - f(y)12
IIfl12 8 j(jO+~)R
1+2s dy dx ::::;:;
Ix-yl
IlfllLP (jo-~)R
Ilfl12 8 j(jO+~)R
::::;Ilfll:;'
Lp
()o-~)R
If(x) IP dx,
If(x)IP dx.
On the other hand, by Sobolev-imbedding theory,
.1
s.l
H ((Jo - "2)R, (Jo
+ "2)R)
C
LP((Jo
.1.1
- "2)R, (Jo
+ "2)R)
for s 2:: ~ - ~, so there is a positive constant k = k(R,p, s) such that
IIfIlH8((jo-~)R,(jo+~)R)
2: kllfIILP((jo-~)R,(jo+~)R)'
whence,
and thus
(jO+~)R
j (jo-~)R
as advertised in the statement
If(x)IP dx 2:: 1] = (
k211fliP
2 LP)
IIfllH
----'?.p-2 ,
8
of the lemma. 0
Below is an improvement of Lion's Lemma 2.4 that will be used in the present, more
general context.
28
6.4. Let p > 2 and s 2: ~ - ~. Suppose {Un}n;:::1 is a bounded sequence in HS•
there is an R > 0 for which
LEMMA
Y+R
lim sup
n-+oo yElR
l
y-R
2
lun(x)1
If
dx = 0,
then it follows that
Proof. We argue by contradiction. If lun(x)IP dx --t 0 is not true, then there must be a
subsequence of {Un}n;:::l, still denoted by {un}n;:::I, and an A> a such that
lun(x)IP dx 2:
A. Then, by Lemma 6.3, there exist, for each n, real numbers Yn and 1] > a such that
J
Note that
1]
J
depends only on R, sand p. It follows from the Holder inequality that
By the Sobolev-imbedding
theorem,
HS ((Yn - R, Yn + R)) c
Lp
((Yn - R, Yn + R)) C L2(p-I) ((Yn - R, Yn + R)),
since p > 2. Hence there exists k = k(R, s,p) such that
If B is an upper bound for
IlunllHs, n = 1,2,···
and consequently
29
, then
which contradicts the assumption.
The lemma is proved. 0
Sketch of a proof of Theorem 6.2. Denote by
J
and µn = Pn(x) dx. Then µn is bounded, so it may be supposed that µn --t µ as n --t 00.
Suppose there is a subsequence of {Pn}n~I, still denoted by {Pn}n~I, which satisfies either
Vanishing or Dichotomy. If Vanishing occurs, then, as before,
lim sup
n-+oo yEIR
1
Iun(x)
2
1
dx = O.
Ix-yl:<:;R
By lemma 6.4
If Dichotomy occurs, there is a µ E (0, µ) such that for any
E
>
0, there corresponds
an no and functions P~, P; E LI(IR), P~, P; 2: 0 such that for n 2: no,
IIPn - (p~ + p;JIIL
1
::::;E,
11p~(x)dx-µISE,
11p~(x)dx-(µ-µ)I::::;E,
and
supp P~ n supp P~ = f/J and dist{supp P~, supp p~} --t
00
as n --t
00.
As before, it may be assumed that
supp P~ C (Yn - Ro, Yn
+ Ro)
and
supp P~ C (-00, Yn - 2Rn) U (Yn
for some fixed Ro > 0, and real sequences {Yn}n~I,
have
{Rn}n~I
with Rn --t
+ 2Rn,
00.
(0)
Then, we
which is to say,
(6.5)
To construct the splitting functions u~ and u; of {Un}n~I
with a ::::;( ::::;1 be such that 0 S ((x) ::::;1 for all x,
((x) = 1 when
for n = 1,2, ... , let ( E
((x) = a when
Ixl S 1,
30
Jxl
2: 2,
ego
and define
1/;(x) = 1- ((x).
For RI > Ro sufficiently large, and for n = 1,2, ... , define
and set
1]~ (x)
1]; (x)
=
=
{
{
(( x - Yn )
R1
(X -
(
Yn )
Rn
'
otherwise,
0,
(( x - Yn )
R1
(X -
(
Yn )
Rn
otherwise.
0,
It follows that for any x E lR,
and
C (Yn - 2Rn, Yn - R1),
supp
1]~
SUpp
1]; C (Yn + RI, Yn + 2Rn),
SUpp (n C (Yn - 2RI, Yn
+ 2RI),
SUpp 1/;n C (-00, Yk - 2Rn) U (Yn
The functions
{Un}n;:;:1 can then be decomposed
Notice that
31
as
+ 2Rn,
(0).
as
E --t
0,
2Rn la~-j1]~(X
+ Yn)atun(X
1
dx
Ix - RII28
Rl
{
::;max
xElR
2
+ Yn)1
la~-j1]~(X
I
X
_
R"
+ Yn)!2}
28
1
=O(E)
32
12
Ilunl Hi(Yn+Rl,Yn+2Rn)
as t
--t
0, and, similarly,
as t
--t
O. Combining these last three estimates and using (6.5) once more yields
(6.6)
as t
--t
O. By the same argument, it appears that
(6.7)
as t
--t
O. If u~ = (nun,
u~ = 'l/Jnun and
Wn
= 1]~Un
+ 1]~un'
then it transpires that
and
supp u~ C (-00, Yn - 2Rn) U (Yn
Using (6.6) and (6.7), it is deduced that
as t
--t
O. Then for large n and small t, we have
33
+ 2Rn,
(0).
Ilu~1I1rk --t µo
and so there is µo E [0, µ] such that
and
Ilu~111rk--t
µ - µo·
From this information, we may derive the contradiction 0 > 8(>') > 8(>'), as before.
Thus Dichotomy is ruled out. By simply adapting the details of the proof of Theorems 2.4
or 4.2, the existence portion of theorem 6.3 is proved.
To prove the decay part of the theorem, we just need to determine the inverse Fourier
transform of (c+ ~;=I ajl~12rj)-I.
Similar calculation as in previous sections leads to the
result
whence
lim
X-+±OO
for some constant
THEOREM
IxI2ro+1k(x)
C. The theorem is complete.
6.5. If the parameters c and
(1) rk > ~,
(2) minx 2:: 0 { c + ~ ajx2rj}
aj
=
C,
0
satisfy
> 0,
(3) ~aj(sinrj7r)yrj
2: 0 for all y 2: 0, and
(4) for any p 2: a and 0 ::::;() ::::;~,
then
(1) the equation (5.1) has a non-trivial solitary-wave solution ¢ = ¢c E Hoo(IR), and
¢c may be chosen to be an even, positive function, strictly monotone decreasing
on (0, (0) and such that ¢c and all its derivatives are bounded and continuous
LI-functions.
(2) Moreover, ifro = min{rj : rj fj. Z,j = 1,·" , k}, then ¢c decays at the rate
for some constant C.
Proof. Write (5.1) in the form
1 (c + £)-Icj>P = -1
1
¢ = -1
p-
p-
34
J
k(x - y)cj>P(y)
dy,
(6.8)
where k is defined by the Fourier symbol
Computing as before, it is seen that
and from this representation, it is obvious that k satisfies all three conditions in Lemma
2.1. Hence the existence part of the theorem is in hand. It is straightforward to check that
where ao is the coefficient corresponding to the power ro in the definiton the symbol of £.
Applying the results in Section 3, the decay part of the theorem is thereby concluded. 0
Acknowledgement:
Part of this research was carried out while both authors were visiting The Centre de Mathematiques et de leurs Applications, Ecole Normale Superieure de
Cachan. The research was partially supported by the National Science Foundation, USA.
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