Orbital stability of solitary waves for a shallow water equation

Physica D 157 (2001) 75–89

Orbital stability of solitary waves for a shallow water equation

Adrian Constantin a,∗, Luc Molinet b

a Department of Mathematics, Lund University, Box 118, S-22100 Lund, Swedenb Département de Mathématiques, Institut Galilée, Université Paris 13, 99 Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

Received 10 May 2000; received in revised form 13 March 2001; accepted 18 May 2001Communicated by C.K.R.T. Jones

Abstract

We prove the orbital stability of the solitary waves for a shallow water equation by means of variational methods, consideringa minimization problem with an appropriate constraint. © 2001 Published by Elsevier Science B.V.

Keywords: Orbital stability; Solitary waves; Shallow water equation; Variational methods; Minimization problem

1. Introduction

1.1. The shallow water equation

There are several classical models describing the motion of waves at the free surface of shallow water under theinfluence of gravity. In shallow water, as depth is assumed to be small compared to the horizontal scale, the verticalvelocity component is very small; further, imposing a linear relation between elevation and fluid velocity, the prop-agation of the modeled waves is restricted in one direction and the obtained evolution equation is one-dimensional.Among these models, we find the celebrated Korteweg–de Vries (KdV) equation [21]

ut + 6uux + uxxx = 0, u(0, x) = u0(x), t > 0, x ∈ R. (1.1)

Here and below u(t, x) represents the wave height above a flat bottom, x is proportional to distance in the direction ofpropagation, and t is proportional to elapsed time. Eq. (1.1) admits solitary wave solutions, i.e. solutions of the formu(t, x) = φ(x − ct) which travel with fixed speed c (traveling waves), and that vanish at infinity. The solitary wavesof the KdV equation retain their individuality under interaction and eventually emerge with their original shapesand speeds (see [15]); for this reason they are called solitons. Moreover, KdV is an integrable infinite-dimensionalHamiltonian system (see [23]). However, the KdV equation does not model the occurrence of breaking for waterwaves (under wave breaking we understand, cf. [29], the phenomenon that a wave remains bounded, but its slopebecomes unbounded in finite time). Indeed, as soon as the initial profile u0 ∈ H 1(R), 1 the solutions of (1.1) areglobal in time (cf. [20]), whereas (cf. [29]), some shallow water waves break.

∗ Corresponding author.E-mail address: [email protected] (A. Constantin).

1 Hk(R), k ∈ N, stands for the Sobolev space of functions with derivatives up to order k having finite L2(R) norm.

0167-2789/01/$ – see front matter © 2001 Published by Elsevier Science B.V.PII: S0 1 6 7 -2 7 89 (01 )00298 -6

76 A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

An alternative model for KdV is the regularized long-wave equation [3]

ut + ux + uux − uxxt = 0, u(0, x) = u0(x), t > 0, x ∈ R. (1.2)

Eq. (1.2) has better analytical properties than the KdV model but it is not integrable and numerical work suggests thatits traveling waves are not solitons (see [14]). As any smooth initial profile with nice decay properties develops intoa solution of permanent form for (1.2) cf. [3], the regularized long-wave equation does not model wave breaking.

Whitham [29] suggested the equation

ut + uux +∫R

k0(x − ξ)ux(t, ξ) dξ = 0, u(0, x) = u0(x), t > 0, x ∈ R (1.3)

with the singular kernel k0(x) = (1/2π)∫R((tanh ξ)/ξ)1/2 eiξx dξ , as another alternative model for shallow water

waves and conjectured that it describes the effect of breaking of waves (for a proof, we refer to [9,26]). The numericalcalculations carried out for the Whitham equation (1.3) do not support the hypothesis that soliton interaction occursfor its traveling waves (cf. [14]).

It is intriguing to know if all these properties like solitons, integrability and breaking waves, may be found in asingle equation. Camassa and Holm [4] proposed 2 the following equation:

ut − utxx + 3uux = 2uxuxx + uuxxx, u(0, x) = u0(x), t > 0, x ∈ R. (1.4)

Camassa and Holm [4] gave a physical derivation for (1.4), obtained the associated isospectral problem and foundthat the equation has solitary waves that interact like solitons. Numerical simulations [5] support their results.The study of the associated isospectral problem proves the integrability of (1.4) in the periodic case for a largeclass of initial data [10]. Results on the existence of global solutions and wave breaking 3 were obtained in [8,9],while the existence of weak solutions is addressed in [30,31]. Concerning integrability, let us mention that (cf.[17]) Eq. (1.4) provides a counter-example to the long-lasting Painlevé conjecture on the complete integrability ofnonlinear partial differential equations [1]. Let us also point out that, corresponding to the Lagrangian formulation, 4

Eq. (1.4) is a re-expression of the geodesic flow in the group of compressible diffeomorphisms of the line [24]. Theconnection of the shallow water flow with infinite-dimensional geometry resembles the fact that the Euler equationof hydrodynamics is an expression of the geodesic flow in the group of incompressible diffeomorphisms of R3 [2],and can be used to study for Eq. (1.4) the existence of global solutions as well as the phenomenon of breaking waves(for which the exact blow-up rate can be determined and, in some cases, the blow-up set), cf. [7].

The plentitude of structures tied into the shallow water equation (1.4) explains the many interesting (and physicallyrelevant) phenomena modeled by it. The equation reconciles the properties which were known for different shallowwater models.

1.2. Solitary waves for the shallow water equation

Camassa and Holm [4] found that the solitary waves for Eq. (1.4) are given by

uc(x, t) = c e−|x−ct|, x ∈ R, (1.5)

2 Eq. (1.4) and its bi-Hamiltonian structure were written down earlier by Fokas and Fuchssteiner [16].3 A formal proof that (1.4) has solutions which blow-up in finite time can be found in [5]; in [7], it is proved that the only way singularities can

develop in a classical solution of (1.4) is in the form of breaking waves.4 The spatial (Eulerian) equation (1.4) describes the water motion from the viewpoint of a fixed observer. In material (Lagrangian) coordinates,

one describes the motion as seen from one of the particles of the fluid (the observer follows the fluid).

A. Constantin, L. Molinet / Physica D 157 (2001) 75–89 77

where c > 0 is the speed. They discovered that these peaked solitary waves 5 are solitons. The solitons are notclassical solutions of the shallow water equation, they have to be understood (see [31]) as weak solutions of thisequation due to the fact that one can write (1.4) as a conservation law

ut + uux + ∂xp ∗ [u2 + 12u

2x] = 0, (1.6)

where ∗ stands for convolution with respect to the spatial variable x ∈ R and

p(x) := 12 e−|x|, x ∈ R. (1.7)

In the form (1.6), one can observe a similarity with (1.1) and (1.3).The existence and stability of solitary waves is one of the fundamental qualitative properties of the solutions

of nonlinear wave equations (cf. [27]). Due to the fact that the solitons hardly interact with each other at all, it isreasonable to expect that they are stable. It is the aim of this paper to prove the stability of the solitary waves forthe shallow water equation (1.4) by using a variational approach: we show that the solitary waves are minima ofthe energy with an appropriate constraint, deducing from this their stability. We would like to point out that thestability of the solitons was proved by an entirely different approach in [11]. The present method has the advantageof ensuring that (1.4) satisfies the physical principle asserting that ground states are stable. 6

This paper is organized as follows. In Section 2, we sketch the steps in our proof of the stability of the solitarywaves: for the full details in the justification of the several delicate technical points, see Section 3. For convenience,we collect in Appendix A some results on the shallow water equation that are used throughout the paper.

2. Outline of the variational approach

For the shallow water equation (1.4), we have conservation of energy [4], i.e. if u is a solution of the equation,then the quantity

1

2

∫R

[u2(t, x) + u2x(t, x)] dx

is preserved as long as the solution exists. If it is present, a state of lowest energy (with an appropriate constraint)is called a ground state. A standard principle in physics asserts that ground states are stable.

This consideration suggests to characterize the solitary waves as ground states and to use a variational approachto prove their stability.

Before giving the exact definition, let us briefly explain what kind of stability is to be expected. Eq. (1.4) is awave equation so that its solutions tend to be oscillations which spread out spatially in a quite complicated way andtherefore in general for two nearby initial profiles the resulting solutions do not remain nearby for all future times. 7

The spreading can be counteracted if we deal with special solutions where the wave is concentrated so that it locallyforms peaks. Also, for a typical nonlinear wave, the phase, speed, and amplitude are coupled. If we define the orbitof a function ϕ to be the setO(ϕ) = ϕ(·+x0), x0 ∈ R, a solitary wave of (1.4) is called orbitally stable if profilesnear its orbit remain at all later times near the orbit. To make this precise, consider the following definition.

5 Interestingly (cf. [29]), the solitary waves which are found in shallow water have a symmetrical peaking of the crests with a finite angle there.6 We should also point out that the approach used in [11] is peculiar in that it is apparently limited to this special equation.7 This can be easily seen in the case of Eq. (1.4) by looking at the time evolution of two solitary waves which initially have their peaks located

at the same point. At time zero their distance, measured in L∞(R), equals the difference of the speeds but as time elapses the distance gets closeto the largest of the two speeds as the amplitude of a solitary wave (1.5) equals to its speed (a picture helps).


Definition. Let ϕc be the profile of the solitary wave of (1.4) traveling with speed c > 0. Then ϕc is orbitally stableif for all ε > 0, there is a δ > 0 such that for any u0 ∈ H 3(R) with ‖u0 − ϕc‖H 1(R) ≤ δ and such that u0 − u0,xx

is a positive integrable function with integral less than 4c, the corresponding solution u(t) of (1.4) with u(0) = u0

satisfies

supt≥0

infr∈R

‖u(t) − ϕc(· − r)‖H 1(R) ≤ ε.

In order to explain the definition, we have to clarify some points. First of all, the form of the partial differentialequation (1.4) suggests to define a strong solution as a function in C([0, T ];H 3(R))∩C1([0, T ];H 2(R)) satisfying(1.4) in C([0, T ];L2(R)) for some T > 0 (we deal with a quasilinear hyperbolic equation, cf. [8]). For an initialprofile u0 ∈ H 3(R), there exists a maximal interval on which the strong solution of (1.4) with initial data u(0) = u0

is defined (see Appendix A). Moreover, if in addition u0 − u0,xx ≥ 0 on R, the corresponding solution is global intime and if u0 ∈ H 4(R) we deal with a classical solution, i.e. a C1-function of time and C3-function of the spatialvariable satisfying (1.4) pointwise. Orbital stability of the solitary wave ϕc means that any solution starting closeto ϕc remains close to some translate of the latter at any later time. But the solitary waves of (1.4) are peaked wavesand therefore not strong solutions. At time zero ϕc −∂2

xϕc = 2cδ (here δ is the Dirac measure) so that the hypothesison the function u0 − u0,xx arises quite naturally if we are interested in classical initial profiles approximating ϕc, asL1(R) is isometrically embedded inM(R), the space of Radon measures of finite total mass.

The standard approach to stability problems is to linearize, as it is commonly assumed that stability is governedby the linearized equation. If the evolution equation is weakly nonlinear, this idea is conclusive for orbital stabilityof the solitary waves for a large variety of equations (see [27]). A different approach 8 consists in proving directlythat the solitary waves are minima of constrained energy, provided that this minimum is unique up to translations.Without uniqueness, one only gets the stability of the set of minima. To clarify whether the ground states are uniqueup to translations or not is a delicate matter and for this reason some results on orbital stability concern the stabilityof the set of ground states 9 and not the stability of the ground states themselves (see [12,13]). In the present model(1.4), the nonlinearity plays a dominant role rather than being a higher-order correction. For this reason, we expectthat in our case the passage from the linear to the nonlinear theory is not an easy task, and may even be false.Our approach is to prove directly that the solitary waves are minima of the energy with an appropriate constraint,deducing from this their stability.

We consider the variational problem

I := infu∈H 1(R)

∫R(u2 + u2

x)∫R(u3 + uu2

x)= 4

3

. (2.1)

In Section 3, we will prove that the above minimization problem has, up to translation, a unique solution ϕ(x) =e−|x|, x ∈ R. We call ground states the family of translates of the function ϕ, x → e−|x−x0| for some x0 ∈ R, asthey are the states of lowest energy.

If the ground state ϕ is not orbitally stable, there are initial profiles un0 ∈ H 3(R) satisfying

un0 − un

0,xx ≥ 0 on R with∫R

(un0 − un

0,xx) ≤ 4,

that approximate ϕ in H 1(R) such that for the corresponding global solutions un(t) of (1.4), we can find positive

8 Used for the ground states of the nonlinear Schrödinger equation [6].9 Set which might be larger than the family of translates of a solitary wave.


times tn ≥ 0 with

infr∈R

‖un(tn, x) − ϕ(x − r)‖H 1(R) ≥ ε, n ≥ 1 (2.2)

for some ε > 0.Note that (see Theorem A.1)

E(u) :=∫R

u2 + u2x, F (u) :=

∫R

u3 + uu2x, u ∈ H 1(R) (2.3)

are conservation laws for the shallow water equation (1.4). From un0 → ϕ in H 1(R), we deduce that

F(un(tn, ·)) = F(un0) → F(ϕ) = 4

3 ,

while

E(un(tn, ·)) = E(un0) → E(ϕ) = 2.

The above relations imply that the sequence

vn(x) := αnun(tn, x), x ∈ R with 3α3

n

∫R

([un(tn, x)]3 + un(tn, x)[u

nx(tn, x)]

2) dx = 4, (2.4)

defined for n ≥ 1, is a minimizing sequence for our variational problem (2.1). Using the principle of concentratedcompactness, we prove that there is a subsequence v3

nk+ vnk

v2nk,x

of v3n + vnv

2n,xn≥1 and suitable points xnk

in order that the translated measures

ρnk:= v3

nk(· + xnk

) + vnk(· + xnk

)v2nk,x

(· + xnk) (2.5)

are such that for every ε > 0, each measure ρnkhas mass less than ε outside some fixed ball centered at zero.

As wnk := vnk

(· + xnk) is also a minimizing sequence for the variational problem (2.1), we can extract a

subsequence of it converging weakly in H 1(R) to some function w ∈ H 1(R). The information on the concentrationof the measures ρnk

combined with the method of compensation-compactness enables us to show that w satisfiesthe constraint. This forces the subsequence to converge actually strongly to w in H 1(R) and w is a solution of ourvariational problem. Therefore w is a ground state. This is in contradiction with relation (2.2) and hence the groundstate ϕ must be orbitally stable.

The full details and the justification of the technical points in the above approach are given in Section 3.We considered the solitary wave (x, t) → e−|x−t | for convenience. The same method (modulo an appropriate

rescaling of the variational problem) applies to the solitary wave (x, t) → c e−|x−ct|, c > 0.

3. Stability of the solitary waves

In this section, we prove (the precise definition of orbital stability of a solitary wave for (1.4) is given in Section2) the following theorem.

Theorem 3.1. The solitary waves of the shallow water equation (1.4) are orbitally stable.

For the reader’s convenience, we first recall the principle of concentration-compactness and the results from thetheory of compensated compactness that will be needed as well as a lower semicontinuity result for variationalintegrals suited for our approach.


Lemma 3.2 (Concentration-compactness [22]). Suppose µnn≥0 is a sequence of probability measures on R :µn ≥ 0,

∫R

dµn = 1. There is a subsequence µnk such that one of the following three conditions holds:

1. (Concentration) There exists a sequence xnk ⊂ R such that for any ε > 0, there is a radius R > 0 with the

property that∫|x−xnk |<R

dµnk(x) ≥ 1 − ε

for all nk .2. (Vanishing) For all R > 0, there holds

limnk→∞

(supx∈R

∫|y−x|<R

dµnk(y)

)= 0.

3. (Dichotomy) There exists a number λ ∈ (0, 1) such that for any ε > 0, there is a number R > 0 and a sequencexnk

with the following property: given R′ > R there are non-negative measures µ1nk, µ2

nk, such that

0 ≤ µ1nk

+ µ2nk

≤ µnk, supp(µ1

nk) ⊂ |x − xnk

| < R, supp(µ2nk) ⊂ |x − xnk

| > R′,

lim supnk→∞

(∣∣∣∣λ −∫R

dµ1nk

∣∣∣∣+∣∣∣∣(1 − λ) −

∫R

dµ2nk

∣∣∣∣)

≤ ε.

The theory of compensated compactness enables one to prove continuity of nonlinear expressions under weakconvergence if appropriate conditions hold (see [18,28]). We will need a specific result in this direction. Beforestating the result, we remind some definitions.

Given q > 1, let

W−1q,loc(R) = v ∈ D′(R) : ηv ∈ W−1

q (R) for all η ∈ D(R),

where D(R) are the test functions, D′(R) the space of all distributions, for r ≥ 1, W 1r (R) the space of all

Lr(R)-functions with distributional derivative in Lr(R), and for q > 1, W−1q (R) the set of distributions w which

can be written in the form w = w0 + ∂xw1 where w0, w1 ∈ Lq(R) (the dual of W 1r (R) where (1/r) + 1/q = 1).

The weak topology on the space of measuresM(R) is defined as follows: µn µ iff∫Rη dµn → ∫

Rη dµ for all

continuous η : R→ R with compact support.

Lemma 3.3 (Hörmander [18]). Let unn≥1 and vnn≥1 be sequences of functions in L2loc(R) such that un

u, vn v in L2loc(R) (weakly) and dun → du, dvn → dv in W−1

2,loc(R). Then unvn uv in the weak topology ofmeasures.

To verify the hypotheses of Lemma 3.3, the following result of Murat will be useful.

Lemma 3.4 (Murat [25]). If E is a bounded subset ofM(R) which is also bounded in the space W−1q,loc(R) for some

q > 2, then E is relatively compact in the space W−12,loc(R).

A particular case of a classical weak lower semicontinuity theorem of Tonelli is also needed.

Lemma 3.5 (Struwe [28]). Let G : R × R → R be a continuous non-negative function with G(u, ·) convex forevery u ∈ R. Then, if un, u ∈ W

1,1loc (R) and un → u in L1(Ω), un,x ux weakly in L1(Ω) for all bounded

Ω ⊂ R, it follows that

A. Constantin, L. Molinet / Physica D 157 (2001) 75–89 81∫R

G(u, ux) dx ≤ lim infn→∞

∫R

G(un, un,x) dx.

Our first result is as follows.

Proposition 3.6. The solutions of the variational problem (2.1) are exactly all translates of ϕ, where ϕ(x) :=e−|x|, x ∈ R.

Proof. Recall the notation introduced in (2.3).To f ∈ H 1(R), we associate the L2(R)-function

g(x) :=

f (x) − fx(x), x < x0,

f (x) + fx(x), x ≥ x0,(3.1)

where x0 ∈ R is some point where the function f takes its maximal value.A straightforward calculation shows that∫R

(f 2(x) + f 2x (x)) dx = 2f 2(x0) +

∫R

g2(x) dx,

∫R

(f 3(x) + f (x)f 2x (x)) dx = 4

3f 3(x0) +

∫R

f (x)g2(x) dx. (3.2)

Let us first show that if an H 1(R)-function f satisfying the constraint is such that f (x0) < 1, then E(f ) > 2.Choose ε ∈ (0, 1) such that f (x0) = 1 − ε (note that ε ≥ 1 forces f to be non-positive and as such it cannot

satisfy the constraint). As F(f ) = 43 ,∫

R

f (x)g2(x) dx = 4ε(1 − ε) + 4

3ε3,

by the second relation in (3.2). But f (x0) is a maximum, so that∫R

f (x)g2(x) dx ≤ (1 − ε)

∫R

g2(x) dx

and from the above calculation, we obtain

4ε + 4ε3

3(1 − ε)≤∫R

g2(x) dx.

From this, f 2(x0) = (1 − ε)2 and the first identity in (3.2), we infer

E(f ) ≥ 2 + 2ε2 + 4ε3

3(1 − ε)> 2.

As E(ϕ) = 2, this proves that in order to solve the variational problem (2.1), it is enough to restrict ourselves to thecase f (x0) ≥ 1. Assuming that there exists such a function f with E(f ) < 2, we obtain a contradiction with thefirst relation in (3.2). Therefore I = 2 and ϕ is a solution of (2.1).

If f is another minimizing function we must have f (x0) ≥ 1 by the above discussion. Then the first relationin (3.2) forces g ≡ 0 and f (x0) = 1, as I = 2. But this implies f (x) = e−|x−x0|, x ∈ R, and the proof iscomplete.

We have now all necessary means to proceed with the following proof.


Proof of Theorem 3.1. Assuming instability, we will show that this leads to a contradiction.If ϕ is not orbitally stable, there is an ε > 0 and a sequence un

0n≥1 in H 3(R) with

un0 − un

0,xx ≥ 0 on R,

∫R

(un0 − un

0,xx) ≤ 4,

and

un0 → ϕ in H 1(R),

such that for the corresponding global solutions un(t) of (1.4), we can find positive times tn ≥ 0 with

infr∈R

‖un(tn, x) − ϕ(x − r)‖H 1(R) ≥ ε, n ≥ 1.

Since un0 → ϕ in H 1(R), we have E(un

0) → E(ϕ) = 2 as n → ∞ and

limn→∞E(un(tn, ·)) = 2

as E is a conservation law for Eq. (1.4), cf. Theorem A.1. From un0 → ϕ in H 1(R), we infer un

0 → ϕ in L∞(R)

and ((un0)

2 + (∂xun0)

2) → ϕ2 + ϕ2x in L1(R) so that F(un

0) → F(ϕ) = 43 . The fact that F is also a conservation

law for (1.4), cf. Theorem A.1, implies now that

limn→∞F(un(tn, ·)) = 4

3 .

Defining the sequence vnn≥1 of H 1(R)-functions as in (2.4), the previous relations read

F(vn) = 43 , n ≥ 1, (3.3)

and

E(vn) → 2, n ≥ 1, (3.4)

so that vnn≥1 is a minimizing sequence for the variational problem (2.1).Using relation (A.2) and Lemma A.3(a) from Appendix A, we deduce that all un(tn, ·) and therefore all vn are

non-negative functions.We proceed now in several steps toward the conclusion.Step 1. There are suitable points xnk

∈ R such that for any ε > 0 there is a radius R > 0 with the property that∫|x|<R

dρnk(x) ≥ 1 − ε,

where ρnkare measures defined by (2.5).

We will prove this claim by using the concentration-compactness principle for the probability measures µn =34 (v

3n + vnv

2n,x), n ≥ 1.

1. Let us first show that vanishing is not possible.Otherwise, we would have

limn→∞

(supx∈R

∫|x−y|<1/2

dµnk(y)

)= 0 (3.5)

for some subsequence µnk.


Note that for f ∈ H 1(R) non-negative,

1

2

∫ x+1/2

x−1/2(f 3 + f f 2

x) ≥∫ x+1/2

x−1/2f 2|fx | ≥ 1

3

(sup

|x−y|<1/2f 3(y) − inf

|x−y|<1/2f 3(y)

),

and

1

2

∫ x+1/2

x−1/2(f 3 + f f 2

x) ≥ 1

2inf

|x−y|<1/2f 3(y),

so that∫ x+1/2

x−1/2(f 3 + f f 2

x) ≥ 1

3sup

|x−y|<1/2f 3(y).

The above inequality yields that for all x ∈ R,∫|x−y|<1/2

dµnk(y) = 3

4

∫|x−y|<1/2

(v3nk

+ vnkv2nk,x

) dy

≤ 3

(∫|x−y|<1/2

(v3nk

+ vnkv2nk,x

) dy

)1/3 ∫|x−y|<1/2

(v2nk

+ v2nk,x

) dy.

Sum over x ∈ Z to get

1 =∫R

dµnk≤ 3

(supx∈Z

∫|x−y|<1/2

dµnk(y)

)1/3 ∫R

(v2nk

+ v2nk,x

) dy.

As vnk is a bounded sequence in H 1(R), being a minimizing sequence for the variational problem (2.1), the

right-hand side of the above inequality tends to zero as n → ∞ by (3.5) and we obtained a contradiction.2. We now rule out dichotomy.

Assume that situation of Lemma 3.2(3) holds.Choosing a sequence εm → 0, corresponding Rm > 0, and passing to a subsequence of µnn≥1, we can achieve

that

supp(µ1m) ⊂ |x − xm| < Rm, supp(µ2

m) ⊂ |x − xm| > 2Rm, (3.6)

and

lim supm→∞

(∣∣∣∣λ −∫R

dµ1m

∣∣∣∣+∣∣∣∣(1 − λ) −

∫R

dµ2m

∣∣∣∣)

= 0. (3.7)

Moreover, we may suppose that Rm → ∞ as m → ∞.Choose Φ ∈ C1(R, [0, 1]) such that Φ(x) = 1 for |x| ≤ 1, Φ(x) = 0 for |x| ≥ 2 and 0 < Φ(x) < 1 for

1 < |x| < 2, with

|Φx(x)| + |Φx(x)|√1 − Φ2(x)

≤ K, |x| > 1 (3.8)

for some K > 0. It suffices to take Φ(x) = 1 − (|x| − 1)6 for x close to ±1 with |x| > 1 (we require (3.8) as below√1 − Φ2 ∈ C1(R) is needed).


Let Φm(x) := Φ((x − xm)/Rm) and define

v1m = vmΦm, v2

m = vm

√1 − Φ2

m.

A straightforward calculation shows

E(vm) = E(v1m) + E(v2

m) −∫R

(vm)2Φ2m,x −

∫R

(vm)2(∂x

√1 − Φ2

m(x)

)2

. (3.9)

Note that by (3.8) and the way we defined Φm, we have

|Φm,x(x)| +∣∣∣∣∂x√

1 − Φ2m(x)

∣∣∣∣ ≤ K

Rm

, x ∈ R.

As vm is a minimizing sequence for the variational problem (2.1), it must be bounded in L2(R). But Rm → ∞for m → ∞ so that the above estimate yields

limm→∞

(∣∣∣∣∫R

(vm)2Φ2m,x

∣∣∣∣+∣∣∣∣∣∫R

(vm)2(∂x

√1 − Φ2

m(x)

)2∣∣∣∣∣)

= 0.

We proved that

limm→∞E(vm) = lim

m→∞(E(v1m) + E(v2

m)). (3.10)

Note that the non-negative functions v1m, v2

m were chosen such that

v1m ≡ vm on |x − xm| < Rm, v2

m ≡ vm on |x − xm| > 2Rm,and recall that 0 ≤ µ1

m + µ2m ≤ µm. Using (3.6) and (3.7), we obtain

lim supm→∞

3

4

∫R

(v1m)3 + v1

m(v1m,x)

2 ≥ lim supm→∞

3

4

∫ xm+Rm

xm−Rm

(v1m)3 + v1

m(v1m,x)

2

= lim supm→∞

3

4

∫ xm+Rm

xm−Rm

(vm)3 + vm(vm,x)2 = lim sup

m→∞

∫|x−xm|<Rm

dµm≥lim supm→∞

∫|x−xm|<Rm

dµ1m = λ. (3.11)

A similar procedure leads to

lim supm→∞

3

4

∫R

(v2m)3 + v2

m(v2m,x)

2 ≥ lim supm→∞

3

4

∫|x−xm|>2Rm

(vm)3 + vm(vm,x)2

= lim supm→∞

∫|x−xm|>2Rm

dµm ≥ lim supm→∞

∫|x−xm|>2Rm

dµ2m = 1 − λ. (3.12)

Clearly v1m, v2

m ∈ H 1(R) so that the renormalized functions

x → v1m(x)

3√(3/4)F (v1

m), x → v2

m(x)

3√(3/4)F (v2

m),

satisfy the constraint of the variational problem (2.1) — recall definition (2.3). Therefore

E

(v1m(x)

3√(3/4)F (v1

m)

)≥ 2, E

(v2m(x)

3√(3/4)F (v2

m)

)≥ 2,


and as the functional E is homogeneous of degree 2, we infer from the above and from (3.11) and (3.12) that forevery ε ∈ (0,minλ, 1 − λ), we have

E(v1m) ≥ 2( 3

4F(v1m))2/3 ≥ 2(λ − ε)2/3, E(v2

m) ≥ 2( 34F(v2

m))2/3 ≥ 2(1 − λ − ε)2/3

for all m large enough. Combining (3.10) with the above two estimates and recalling that vm is a minimizingsequence for the variational problem (2.1), we deduce that for all ε > 0 small enough

2 = limm→∞E(vm) = lim

m→∞(E(v1m) + E(v2

m)) ≥ 2((λ − ε)2/3 + (1 − λ − ε)2/3).

But for 0 < λ < 1, we have λ2/3 + (1 − λ)2/3 > 1. Letting ε → 0 in the previous inequality, we obtain the desiredcontradiction that rules out dichotomy.

As vanishing and dichotomy do not hold, Step 1 is finished.Step 2. There is a subsequence of wnk

:= vnk(· + xnk

) converging weakly in H 1(R) to some w satisfying theconstraint in (2.1), if the translates are chosen such that the assertion from Step 1 holds.

As the above-defined wnk is a minimizing sequence for the variational problem (2.1), we can extract a sub-

sequence, denoted for the sake of simplicity again by wnk, such that wnk

converges weakly in H 1(R) to somefunction w ∈ H 1(R). Using the method of compensated compactness we will prove that

∫R(w3 + ww2

x) = 43 .

First of all, observe that, just like the functions vnk, all wnk

are non-negative. Applying Lemma 3.5 with

G : R× R→ R, G(x, y) =

x3 + xy2, x ≥ 0, y ∈ R,0, x ≤ 0, y ∈ R,

we deduce that∫R

(w3 + ww2x) ≤ lim inf

nk→∞

∫R

(w3nk

+ wnkw2

nk,x) = 4

3(3.13)

(the hypotheses of Lemma 3.5 can easily be checked by standard Sobolev imbeddings).Let us first prove that

wnk(∂xwnk

) wwx weakly in L2(R). (3.14)

As the sequence wnk is bounded in H 1(R), we deduce that it is also bounded in L∞(R) and that it converges

pointwise a.e. (almost everywhere) to w. Therefore, the sequence w2nk

is bounded in H 1(R) and it converges a.e.to w2, which forces w2

nk w2 in H 1(R). This on its turn implies (3.14).

We will show now that there are subsequences of ∂xwnk, wnk

(∂xwnk), denoted for simplicity by the same

subscripts, such that

wnk(∂xwnk

)2 → ww2x in D′(R). (3.15)

Note that wnk= unk (tnk

, · + xnk) and let us denote by ‖µ‖M the total variation of a Radon measure µ on R. Using

Lemma A.3, we deduce that

‖∂2xwnk

‖M ≤ ‖(1 − ∂2x )u

nk (tnk, · + xnk

)‖L1(R) + ‖unk (tnk, · + xnk

)‖L1(R)

= ‖(1 − ∂2x )u

nk (0, · + xnk)‖L1(R) + ‖unk (0, · + xnk

)‖L1(R) = 2‖unk

0 (· + xnk) − ∂2

xunk

0 (· + xnk)‖L1(R) ≤ 8,

recalling the conditions on the initial profiles un0n≥1 (see the beginning of the proof). This means that

∂2xwnk

is bounded in M(R). (3.16)


Clearly ∂x[wnk(∂xwnk

)] = (∂xwnk)2 +wnk

(∂2xwnk

). The L1(R)-norm of (∂xwnk)2 and the L∞(R)-norm of wnk

are bounded as wnk

is a minimizing sequence for (2.1). Together with (3.16), this ensures that

∂x[wnk(∂xwnk

)] is bounded in M(R). (3.17)

Let us show now that ∂2xwnk

and ∂x[wnk(∂xwnk

)] are bounded in the space W−1∞,loc(R). For this, pick f ∈ D(R)

and g ∈ W 11 (R). Then, using the definition of the duality pairing in W−1∞ (R) × W 1

1 (R), we have

〈f (∂2xwnk

), g〉 =∫R

fg(∂2xwnk

)

= −∫R

(fxg + fgx)(∂xwnk) ≤ C1(f )‖∂xwnk

‖L∞(R)‖g‖W 11 (R)

≤ C2(f )‖g‖W 11 (R)

using Lemma A.4 and the boundedness of wnk in L∞(R). This shows that

∂2xwnk

is bounded in W−1∞,loc(R). (3.18)

The boundedness of ∂xwnk and wnk

in L∞(R) (already explained) and a calculation along the same lines as theone we just did ensure that

∂x[wnk(∂xwnk

)] is bounded in W−1∞,loc(R). (3.19)

In view of (3.16)–(3.19) and Lemma 3.4, we have that both sequences ∂2xwnk

and ∂x[wnk(∂xwnk

)] are relativelycompact in the space W−1

2,loc(R). As wnk w weakly in L2(R) and (3.14) holds, we deduce relation (3.15) by

Lemma 3.3.We already proved that the sequences wnk

and ∂xwnk are bounded in L∞(R) so that wnk

(∂xwnk)2 is bounded

in L2(R), using the information that wnk is a minimizing sequence for our variational problem. We deduce from

here that wnk(∂xwnk

)2 must have a subsequence converging weakly in L2(R) and relation (3.15) enables us toidentify the limit as ww2

x .As wnk

w in H 1(R), a Sobolev imbedding theorem ensures that

w3nk

→ w3 in L2loc(R).

In conclusion, we know that

w3nk

+ wnk(∂xwnk

)2 w3 + ww2x in L2

loc(R). (3.20)

In view of Step 1 for every ε > 0, there is some Rε > 0 such that

3

4

∫|x|<Rε

(w3nk

+ wnkw2

nk,x) > 1 − ε

for all nk large enough. Using (3.20) and considering the L2(R)-scalar product with the characteristic function ofthe set |x| < Rε, we obtain∫

|x|<Rε

(w3 + ww2x) = lim

nk→∞

∫|x|<Rε

(w3nk

+ wnkw2

nk,x) ≥ 4

3− 4

3ε.

By the arbitrariness of ε > 0 and the fact that all functions wnkare non-negative, we conclude that∫

R

(w3 + ww2x) ≥ 4

3.


Together with (3.13), this shows that in the above relation we actually have equality and we are done with the proofof Step 2.

Step 3. The sequence wnk found in Step 2 converges strongly to w ∈ H 1(R) which is a solution of our variational

problem.By Lemma 3.5, we have

E(w) ≤ lim infnk→∞E(wnk

) = 2

with the notation introduced in (2.3). Step 2 guarantees that w ∈ H 1(R) satisfies the constraint so that w is a solutionof (2.1).

A simple calculation shows that

‖w − wnk‖2H 1(R)

= E(wnk) − 2

∫R

wwnk− 2

∫R

wxwnk,x + E(w).

But E(w) = 2 and wnk w in H 1(R) together with the fact that wnk

is a minimizing sequence for our variationalproblem imply that the right-hand of side of the above identity tends to zero as nk → ∞.

Step 4. The contradiction with the assumption of instability.From Step 3 and Proposition 3.6, we know that w is some translate of ϕ. Recalling what wnk

stands for, we have

limnk→∞‖unk (tnk

, · + xnk) − ϕ(· − r)‖H 1(R) = 0

for some r ∈ R. The translation invariance of the H 1(R)-norm enables us to move xnkto the ground state ϕ in the

above relation and in the new form the incompatibility with (2.2) is obvious.

Acknowledgements

We thank Prof. W.A. Strauss for many useful discussions. We are also grateful to Prof. C.K.R.T. Jones and thereferees for suggestions and comments.

Appendix A

For the reader’s convenience, we summarize briefly some results about Eq. (1.4) that were used in the previoustwo sections.

Associating to a solution of (1.4) the potential y := u − uxx, one can write Eq. (1.4) in the following equivalentform:

yt + yxu + 2yux = 0, y(0, x) = y0(x), t > 0, x ∈ R, (A.1)

a quasilinear evolution equation of hyperbolic type

dy

dt+ A(y)y = 0 in L2(R), y(0) = y0, t > 0,

where, for y ∈ H 1(R), A(y) is the operator

A(y) := ((1 − ∂2x )

−1y)∂x + 2((1 − ∂2x )

−1y)xId

with domain v ∈ L2(R) : ((1 − ∂2x )

−1y)v ∈ H 1(R).


Eq. (A.1) can be analyzed with Kato’s semigroup approach to the Cauchy problem for quasilinear hyperbolicevolution equations [19] in order to obtain the following well-posedness result.

Theorem A.1 (Constantin and Escher [8]). Given an initial data u0 ∈ H 3(R), there exists a maximal timeT = T (u0) > 0 so that, on [0, T ), Eq. (1.4) admits a unique solution u = u(·, u0) ∈ C([0, T );H 3(R)) ∩C1([0, T );H 2(R)). Moreover,∫

R

[u2(t, x) + u2x(t, x)] dx and

∫R

[u3(t, x) + u(t, x)u2x(t, x)] dx

are conserved on [0, T ) and the solution depends continuously on the initial data. Further, if T < ∞, thenlim supt↑T ‖u(t)‖H 3(R) = ∞. If u0 ∈ H 4(R), then the solution possesses additional regularity, u ∈ C([0, T );H 4(R)) ∩ C1([0, T );H 3(R)) and is a classical solution.

Regarding the existence of global solutions, we have the following theorem.

Theorem A.2 (Constantin and Escher [8]). Assume that u0 ∈ H 3(R) is such that y0 := u0 − u0,xx ∈ L1(R) and isnon-negative. Then the solution u(t) to (1.4) with initial data u0 exists globally in time.

We will also need the following lemma.

Lemma A.3 (Constantin and Escher [8]). Let u0 ∈ H 3(R) and assume that y0 = u0 − u0,xx is non-negative andbelongs to L1(R). If u(t) denotes the corresponding global solution in H 3(R) to (1.4), we have for all t ≥ 0 that

y(t, ·) := u(t, ·) − uxx(t, ·) ≥ 0 on R, ‖u(t, ·)‖L1(R) = ‖y(t, ·)‖L1(R) = ‖y0‖L1(R).

Lemma A.4. If u0 ∈ H 3(R) is such that u0 − u0,xx is non-negative and belongs to L1(R), then

u2x(t, x) ≤ u2(t, x), t ≥ 0, x ∈ R,

where u(t) denotes the global solution of (1.4) with initial data u0.

Proof. By Lemma A.3, we have that

y(t, x) := u(t, x) − uxx(t, x) ≥ 0, t ≥ 0, x ∈ R.

Note that u(t, x) is given by the convolution u(t, x) = p ∗ y with p(x) := 12 e−|x|, x ∈ R, and therefore

u(t, x) = 1

2e−x

∫ x

−∞eξ y(t, ξ) dξ + 1

2ex∫ ∞

x

e−ξ y(t, ξ) dξ, t ∈ [0, T ), x ∈ R, (A.2)

from where we infer

ux(t, x) = −1

2e−x

∫ x

−∞eξ y(t, ξ) dξ + 1

2ex∫ ∞

x

e−ξ y(t, ξ) dξ, t ∈ [0, T ), x ∈ R. (A.3)

As the function y(t, ·) is non-negative for all t ≥ 0, the assertion follows by simply comparing the expressionsobtained by squaring (A.2) and (A.3).


References

[1] M.J. Ablowitz, A. Ramani, H. Segur, Nonlinear evolution equations and ordinary differential equations of Painlevé type, Lett. NuovoCimento 23 (1978) 333–338.

[2] V. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluidesparfaits, Ann. Inst. Fourier (Grenoble) 16 (1966) 319–361.

[3] B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London272 (1972) 47–78.

[4] R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664.[5] R. Camassa, D. Holm, J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33.[6] T. Cazenave, P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85 (1982)

549–561.[7] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier

(Grenoble) 50 (2000) 321–362.[8] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa 26 (1998) 303–328.[9] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229–243.

[10] A. Constantin, H.P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math. 52 (1999) 949–982.[11] A. Constantin, W.A. Strauss, Stability of peakons, Commun. Pure Appl. Math. 53 (2000) 603–610.[12] A. de Bouard, J.C. Saut, Remarks on the stability of generalized KP solitary waves, in: Mathematical Problems in the Theory of Water

Waves, Contemporary Mathematics, Vol. 200, American Mathematical Society, Providence, RI, 1996, pp. 75–84.[13] A. de Bouard, J.C. Saut, Solitary waves of generalized Kadomtsev–Petviashvili equations, Annales IHP Analyse Nonlinéaire 14 (1997)

211–236 .[14] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984.[15] P.G. Drazin, Solitons, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1983.[16] A.S. Fokas, B. Fuchssteiner, Symplectic structures: their Bäcklund transformation and hereditary symmetries, Physica D 4 (1981) 47–66.[17] C. Gilson, A. Pickering, Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations, J. Phys. A 28

(1995) 2871–2888.[18] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer, Berlin, 1997.[19] T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Stud. Appl. Math. 8 (1983) 93–126.[20] C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction

principle, Commun. Pure Appl. Math. 46 (1993) 527–620.[21] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary

waves, Phil. Mag. 39 (1895) 422–443.[22] P.-L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, Annales IHP Analyse

Nonlinéaire 1 (1984) 104–145.[23] H.P. McKean, Integrable systems and algebraic curves, in: Global Analysis, Springer Lecture Notes in Mathematics, Vol. 755, Springer,

Berlin, 1979, pp. 83–200.[24] G. Misiolek, A shallow water equation as a geodesic flow on the Bott–Virasoro group, J. Geom. Phys. 24 (1998) 203–208.[25] F. Murat, Compacité par compensation, Annali Sc. Norm. Sup. Pisa 5 (IV) (1978) 489–507.[26] P. Naumkin, I. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, Vol. 133, Transl. Math. Monographs, Providence, RI,

1994.[27] W.A. Strauss, Nonlinear Wave Equations, Conf. Board Math. Sci., Vol. 73, American Mathematical Society, Providence, RI, 1989.[28] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Berlin, 1990.[29] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1980.[30] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Commun. Pure Appl. Math. 53 (2000) 1411–1433.[31] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000) 45–61.

Orbital stability of solitary waves for a shallow water equation

Documents

Transcript of Orbital stability of solitary waves for a shallow water equation