# On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation

Authors: Lynnyngs Kelly Arruda
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

This paper is concerned with instability of periodic travelling wave solutions of the modified Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental periodL will be constructed by using Jacobi's elliptic functions. It will be shown that these solutions, called dnoidal
waves, are nonlinearly unstable in the energy space for a range of their speeds of propagation and
periods.


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## Body

## 1. Introduction

The original Boussinesq equations are among the classical models for the propagation of small amplitude, planar long waves on the surface of water [1, 2]. These equations possess special travelling wave solutions known as Scott Russel’s solitary waves or solitons [3, 4], cnoidal waves [5], and dnoidal waves ([6], Section 3 below). The cnoidal and dnoidal wave solutions are periodic travelling waves written in terms of the Jacobi elliptic functions.Our purpose is to investigate the nonlinear stability of periodic travelling wave solutionsϕ(x-ct) of the modified Boussinesq equation(1.1)utt-uxx+(u3+uxx)xx=0.
The above equation (1.1), has the following equivalent form as a Hamiltonian system(1.2)ut=vx,vt=(u-uxx-u3)x
for x∈ℝ, t>0. Here subscripts t and x denote partial differentiation with respect to t and x.The above equation conserves energy, namely, the integral(1.3)H(u,v)=12∫0L(u2+v2+ux2-u42)dx
does not depend on the time t. Another conservation law is the momentum(1.4)I(u,v)=∫0Luvdx
which turns out to be a relevant quantity in the investigation of stability properties of travelling waves.To make precise the notion of stability we use, letτs be the translation by s, τsϕ(x)=ϕ(x+s) for x∈ℝ and let ϕ⃗c=(ϕc(x-ct),ψc(x-ct)) be an L-periodic travelling wave solution to system (1.2), where ϕc:ℝ→ℝ, ψc:ℝ→ℝ, L>0 is the period of ϕc and ψc, and c is the wave’s speed of propagation. If we define the ϕ⃗c-orbit to be the set Ωϕ⃗c={ϕ⃗c(·+s),s∈ℝ}, ϕ⃗c is called orbitally stable if profiles near its orbit remain near the orbit for as long as it exists.So, we have the following definition. LetX be a Hilbert space.Definition 1.1 (Orbital Stability).
Letϕ⃗c=(ϕc(x-ct),ψc(x-ct))∈X be an L-periodic travelling wave solution to system (1.2). We say that the orbit Ωϕ⃗c is stable in the X-sense by the flow of system (1.2) if for each ϵ>0 there exists δ=δ(ϵ)>0 such that if u⃗0∈X and infs∈ℝ∥u⃗0-τs(ϕ⃗c)∥X<δ, then the solution u⃗(t) of (1.2) with u⃗(0)=u⃗0 satisfies, for all t for which u⃗=(u,v) exists,
(1.5)infs∈ℝ∥u⃗(t)-τs(ϕ⃗c)∥X<ϵ.
Otherwise, we say that Ωϕ⃗c is X-unstable.Here,X:=Hper1([0,L])×Lper2([0,L]). (The choice of norm in (1.5) is dictated by the form of the Hessian or “linearized Hamiltonian” H′′(ϕ⃗c)+cI′′(ϕ⃗c) and varies from problem to problem.)Inserting theL-periodic travelling wave solution ϕ⃗c=(ϕc(x-ct),ψc(x-ct)) in (1.2) leads to the system (1.6)-cϕc′(ξ)=ψc′(ξ),-cψc′(ξ)=(ϕc-ϕc′′-ϕc3)′(ξ),
where ' connotes d/dξ and ξ=x-ct. Integrating the latter system, we obtain the nonlinear system (1.7)-cϕ(ξ)=ψ(ξ)+K1,-cψ(ξ)=ϕ(ξ)-ϕ′′(ξ)-ϕ3(ξ)+K2,
where K1,K2 are integration constants, which will be considered equal to zero here. Then, we obtain(1.8)(H′+cI′)(ϕ⃗c)=0.
Next observe that relation (1.8) characterizes ϕ⃗c=(ϕc,ψc) as a critical point of H subject to the constraint I(u,v)=I(ϕc,ψc). In order to prove instability for ϕ⃗c, we will examine the relation between the concavity properties of the function(1.9)d(c)=H(ϕ⃗c(·))+cI(ϕ⃗c(·)),
and the properties of the functional H near the critical point ϕ⃗c under the constraint I=constant.Bona and Sachs in [3] proved that the well-known solitary waves ϕ⃗=(ϕc(x-ct),ψc(x-ct)) of the generalized Boussinesq equation(1.10)ut=vx,vt=(u-uxx-up)x
are stable in the H1(ℝ)×L2(ℝ) norm for speeds c such that (p-1)/4<c2<1 if d in (1.9) is a convex function of c. The aim of this paper is to prove that the solutions given by Theorem 3.2 below are unstable if d(c) is concave. The proof follows the main ideas of Liu [4] (see also Bona et al. in [7]). Differently from the solitary wave solutions case, we do not know explicit periodic travelling wave solutions in the x-variable for the system (1.10) for every p. For this reason, we will treat here only the case p=3. Stability of dnoidal waves for this case is also treated by the author in a forthcoming paper [6]. Regarding the classical case p=2, in [5] the author proves nonlinear stability properties of a class of L-periodic travelling wave solutions, called cnoidal waves, in the energy space Hper1([0,L])×Lper2([0,L]), by periodic disturbances with period L.In this paper, we first show the existence of a smooth curvec↦ϕ⃗c=(ϕc,ψc) of dnoidal wave solutions to system (1.2), with a fixed period L (Theorem 3.2 below). Then, a proof of orbital instability of these solutions is established in X for a certain range of their speeds of propagation and periods, based on a modification of the general procedure of [8]. More precisely, our main result regarding stability of the dnoidal waves ϕ⃗c, given by Theorem 3.2 below, is the following.Theorem 1.2 (Instability Theorem).
Letc∈(-1,1) and L>π2. Then the orbit Ωϕ⃗c is X-unstable with respect to the flow of the modified Boussinesq equation, provided c2<1/2 and 1-c2>2π2/L2.The plan of this paper is as follows. A discussion of the evolution equation (1.1) and its natural invariants is given in Section 2. In Section 3 we introduce a smooth family {ϕ⃗c}c on the parameter c, of positive dnoidal wave solutions to system (1.2), with a fixed period L (Theorem 3.2 below) and in Section 4 we present a complete study of the spectrum of the operator ℒc. The existence of the smooth curve c↦ϕ⃗c will allow us to differentiate the function d(c). Then, in section 5, we prove that  d(c)  is indeed concave, for a certain range of speeds and periods of ϕ⃗c, which will imply our result. In Section 6 the Lyapunov functional [8, 9] is constructed and the instability result is proved. In Appendix, we give a review of those results about Jacobian elliptic functions which we use throughout the paper.We remark that orbital instability ofϕ⃗ is established with respect to perturbations of periodic functions of the same period L in X.The following notation will be used:(1.11)〈f,g〉0=〈f,g〉Lper([0,L])2=∫0Lfgdx,〈f,g〉1=〈f,g〉Hper([0,L])1=∫0Lfgdx+∫0Lf′g′dx,∥f∥0=∥f∥Lper([0,L])2=(∫0Lf2dx)1/2,∥f∥1=∥f∥Hper([0,L])1=(∫0Lf2dx+∫0Lf'2dx)1/2,〈(f,g),(u,v)〉=〈(f,g),(u,v)〉Lper([0,L])2×Lper([0,L])2=∫0Lfudx+∫0Lgvdx,∥(f,g)∥=∥(f,g)∥Lper([0,L])2×Lper([0,L])2=(∫0Lf2dx+∫0Lg2dx)1/2,∥(f,g)∥X=∥(f,g)∥Hper([0,L])1×Lper([0,L])2=(∫0Lf2dx+∫0Lf'2dx+∫0Lg2dx)1/2.


## 2. The Evolution Equation

The next lemma is the periodic version of a particular case of [4, Lemma 1.1].Lemma 2.1.
Letu⃗0=(u0,v0)∈X≡Hper1([0,L])×Lper2([0,L]). Then there exist T>0 and a uniquely weak solution u⃗=(u,v) of (1.2) with u⃗(0)=u⃗0.Proof.
In order to obtain the existence of weak solutions for the system (1.2), we consider the approximate problem
(2.1)u⃗t+Au⃗=F(u⃗),u⃗(0)=u⃗0n,
with u⃗0n∈D(A)=Hper3([0,L])×Hper1([0,L]) and u⃗0n→u⃗0 in X, where
(2.2)A=(0-∂x-∂x+∂x30)
and -A is the infinitesimal generator of a C0 group of unitary operators in X and F=F(t,u,v)=(0-∂x(u3)). Since F∈C∞, the map (u,v)→(0,∂x(u3)) is locally Lipschitz on X. But then for all u⃗0n∈D(A), there exists a Tn>0 such that the initial value problem (2.1) has a unique solution u⃗n∈C([0,Tn);D(A))∩C1([0,Tn);X). Moreover, if Tn<∞, then
(2.3)limt→Tn∥u⃗n(t)∥X=∞,
by the semigroup theory [10]. By (2.1), we estimate on [0,Tn)(2.4)12ddt∥u⃗n(t)∥X2=〈u⃗n(t),∂tu⃗n(t)〉X=〈u⃗n,-Au⃗n+F(u⃗n)〉X≤|∫0L∂x(un)3(t)·vn(t)dx|≤∥∂x(un)3(t)∥0∥vn(t)∥0=3(∫0L|(un(t))2(un(t))x|2dx)1/2∥vn(t)∥0≤3∥un(t)∥∞2(∫0L(uxn(t))2dx)1/2∥vn(t)∥0≤3S(L)∥un(t)∥12∥uxn∥0∥vn(t)∥0≤3S(L)∥u⃗n(t)∥X2∥u⃗n∥X2,
where we used in the first equality above that -Au⃗n∈X.
Consider nowf̃(s)=3S(L)s, which is a continuous, positive and increasing function on ℝ+. Then by Gronwall’s inequality, it follows that
(2.5)∥u⃗n(t)∥X2≤∥u⃗0n∥X2exp[∫0tf̃(∥u⃗n(τ)∥X2)dτ],on[0,Tn).
We compare∥u⃗n(t)∥X2 with the maximal solution y(t)(2.6)y(t)≡1(supn∥u⃗0n∥X)21-3St(supn∥u⃗0n∥X)2,t∈[0,T0)≡[0,13S(supn∥u⃗0n∥X)2)
of the scalar Cauchy problem
(2.7)dydt=f̃(y)y,y(0)=y0=supn∥u⃗0n∥X2.
It follows that
(2.8)∥u⃗n(t)∥X2≤y(t),on[0,Tn)∩[0,T0).
LetT<T0. Then u⃗n is defined on [0,T] for all n. Moreover,
(2.9)∥u⃗n(t)∥X2≤C0y0=K2
on [0,T], where K is a constant independent of n, since by (2.5), (2.8), and the fact that y(t) is bounded on [0,T], we have the following inequality on [0,T]:
(2.10)∥u⃗n(t)∥X2≤∥u⃗0n∥X2exp[∫0tf̃(∥u⃗n(τ)∥X2)dτ]≤∥u⃗0n∥X2exp[∫0Tf̃(y(τ))dτ]≤C0(T)∥u⃗0n∥X2≤C0(T)y0.
Finally, from (2.9) and standard weak limit arguments, we have the existence of a unique solution u⃗(t)∈C([0,T];X).Proposition 2.2.
The unique solutionu⃗(t) of (1.2) with initial data u⃗(0)=u⃗0,  which is given by Lemma 2.1,  satisfies
(2.11)H(u⃗(t))=H(u,v)=constant,0,I(u⃗(t))=I(u,v)=constant,0.The proof is elementary.


## 3. Existence of a Smooth Curve of Dnoidal Wave Solutions with a Fixed PeriodL for the System (1.2)

This section is devoted to establish the existence of a smooth curve of periodic travelling wave solutions for the system (1.2), which are solutions of the form(3.1)u⃗(x,t)=(u(x,t),v(x,t))=(ϕ(x-ct),ψ(x-ct)).Substituting (3.1) in (1.2) leads to the system(3.2)-cϕ′(ξ)=ψ′(ξ),-cψ′(ξ)=(ϕ-ϕ′′-ϕ3)′(ξ),
where ' denotes d/dξ and ξ=x-ct. Integrating (3.2), we obtain the nonlinear system(3.3)-cϕ(ξ)=ψ(ξ)+K1,-cψ(ξ)=ϕ(ξ)-ϕ′′(ξ)-ϕ3(ξ)+K2,
where K1,K2 are integration constants, which will be considered equal to zero here. Then, ϕ must satisfy(3.4)ϕ′′-wϕ+ϕ3=0,
where w=w(c)=1-c2 will be considered positive.Next, we show how to construct a smooth curve of solutions for (3.4) with a fixed fundamental period L, and depending on the parameter c. In order to do this, we first observe from (3.4) that ϕ satisfies the first-order equation
(3.5)(ϕ′)2=12[-ϕ4+2wϕ2+4Bϕ]=12(η12-ϕ2)(ϕ2-η22),
where Bϕ is an integration constant and -η1, η1, -η2, η2 are the real zeros of the polynomial pϕ(t)=-t4+2wt2+4Bϕ, which satisfy the relations(3.6)2w=η12+η22,4Bϕ=-η12η22.
Moreover, we assume without lost of generality that η1>η2>0 and we obtain from (3.5) that η2≤ϕ≤η1. By defining φ=ϕ/η1 and k2=(η12-η22)/η12, (3.5) becomes (φ′)2=(η12/2)(1-φ2)(φ2-1+k2). We also impose the crest of the wave to be at ξ=0, that is, ϕ(0)=1. Now, we define a further variable ψ via the relation φ2=1-k2sin2ψ and so we get that (ψ′)2=(η12/2)(1-k2sin2ψ). Then we obtain for l=η1/2 that ∫0ψ(ξ)(dt/1-k2sin2t)=lξ. Therefore, from the definition of the Jacobian elliptic function y=sn(u;k) (see in the appendix or in Byrd and Friedman [11]), we can write the last equality as sinψ=sn(lξ;k) and hence φ(ξ)=1-k2sn2(lξ;k)=dn(lξ;k). Returning to the initial variable, we obtain the called dnoidal wave solution associated to (3.4),(3.7)ϕ(ξ)≡ϕ(ξ;η1,η2)=η1dn(η12ξ;k)
with(3.8)k2=η12-η22η12,η12+η22=2w,0<η2<η1.
Next, dn has fundamental period 2K, dn(u+2K;k)=dn(u;k), where K=K(k) represents the complete elliptic integral of the first kind (see appendix); it follows that the dnoidal wave ϕ in (3.7) has fundamental period, Tϕ, given by(3.9)Tϕ≡22η1K(k).
Now, we show that Tϕ>2π/w. First, we express Tϕ as a function of η2 and w. In fact, for every η2∈(0,w), there is a unique η1∈(w,2w) satisfying the first relation in (3.6), namely, η1=2w-η22. So, from (3.9) we obtain(3.10)Tϕ(η2,w)=222w-η22K(k(η2)),withk2(η2,w)=2w-2η222w-η22.Then, by fixingw>0, we have that Tϕ→+∞ as η2→0 and Tϕ(η2)→(π2/w) as η2→w. So, since the mapping η2↦Tϕw(η2) is strictly decreasing (see proof of Proposition 3.1), it follows that Tϕ>2π/w.Now, we obtain a dnoidal wave solution with periodL. For w0>2π2/L2, there is a unique η2,0∈(0,w0) such that Tϕ(η2,0,w0)=L. So, for η1,0 such that η1,02+η2,02=2w0, the dnoidal wave ϕ(·)=ϕ(·,η1,0,η2,0) has a fundamental period L and satisfies (3.4) with w=w0.By the above analysis the dnoidal waveϕ(·,η1,η2) in (3.7) is completely determined by w and η2 and will be denoted by ϕw(·;η2) or ϕw.The next result, which corresponds to Theorem2.1 and Corollary 2.2 in [12], is concerned with the existence of a smooth curve of dnoidal wave solutions for (3.4).Proposition 3.1.
LetL>0 be arbitrary but fixed. Consider  w0>2π2/L2 and the unique  η2,0=η2(w0)∈(0,w0) such that  Tϕw0=L. Then,   (1)
there exist an intervalℐ(w0) around w0, an interval J(η2,0) around η2,0, and a unique smooth function Λ:ℐ(w0)→J(η2,0) such that Λ(w0)=η2,0 and
(3.11)222w-η22K(k)=L,
where w∈ℐ(w0), η2=Λ(w), and k2=k2(w)∈(0,1) is defined by (3.10);(2)
the positive dnoidal wave solution in (3.7), ϕw(·;η1,η2), determined by η1≡η1(w)=2w-η22, η2≡η2(w), has fundamental period L and satisfies (3.4). Moreover, the mapping w∈ℐ(w0)→ϕw∈Hper1([0,L])is a smooth function;(3)
ℐ(w0) can be chosen as (2π2/L2,+∞);(4)
the mappingΛ:(2π2/L2,+∞)→J(η2,0) is strictly decreasing.Proof.
From this result we conclude the following existence theorem.Theorem 3.2.
LetL>π2.  Then there exists a smooth curve of dnoidal wave solutions for the system (1.2) in Hpern([0,L])×Hperm([0,L]),  n,m≥0 which satisfy the system (3.3) with integration constants K1=K2=0;  this curve is given, for w(c)=1-c2, by
(3.12)c∈(-1-2π2L2,1-2π2L2)→(ϕw(c),ψw(c)).
Moreover,ϕc(ξ):=ϕw(c)(ξ)=2w-η22dn[(2w-η22/2)ξ;k],ψw(c)=-cϕw(c), where the smooth function η2≡η2(w(c)) is given by Proposition 3.1 and k=k(w(c)) by (3.10).Remark 3.3.
∂ϕ⃗c/∂c is in Hper∞([0,L])×Hper∞([0,L]) as soon as in Hper1([0,L])×Hper1([0,L]).This follows from the equation and a bootstrap argument.


## 4. Spectral Analysis

In this section, we study the spectral properties associated to the linear operator(4.1)ℒc=H′′(ϕw(c),ψw(c))+cI′′(ϕw(c),ψw(c))
determined by the periodic solutions (ϕw(c),ψw(c)) found in Theorem 3.2. We compute the Hessian operator ℒc by calculating the associated quadratic form, which is denoted by 𝒬c. By definition, 𝒬c(g,h) is the coefficient of ϵ2 in (4.2)H(ϕw(c)+ϵg,ψw(c)+ϵh)+cI(ϕw(c)+ϵg,ψw(c)+ϵh),
and so is given by (4.3)𝒬c(g,h)=∫0L{12(g2+gx2+h2)-32ϕw(c)2g2+cgh}dx=∫0L{12[(1-c2)g2+g'2-3ϕw(c)2g2]+12(h+cg)2dx}:=𝒬c1(g)+12∥h+cg∥02.
Note that 𝒬c is the sum of the quadratic form 𝒬c1 associated to the operator -(d2/dx2)+1-c2-3ϕw2 and the nonnegative term (1/2)∥h+cg∥02. From (3.2) for the dnoidal wave (ϕw(c),ψw(c)), it follows that g=ϕw(c)′ and h=ψw(c)′ satisfy ℒc(g,h)=0. To see that this is the only eigenfunction corresponding to the eigenvalue zero and the other expected properties of the operator ℒc, we will first consider the following result about the periodic eigenvalue problem:
(4.4)ℒdnξ:=(-d2dx2+w-3ϕw2)ξ=λ,ξ(0)=ξ(L),ξ′(0)=ξ′(L),
where ϕw is given by Proposition 3.1.The following result is a consequence of the Floquet theory (Magnus and Winkler [13]) and can be found in [12].Theorem 4.1.
Letℒdn be the linear operator defined on  Hper2([0,L]) by (4.4). Then the first three eigenvalues  β1,  β2, and  β3 of  ℒdn are simple, and satisfy  β1<0=β2<β3, and  ϕ′ is the eigenfunction of  β2. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues which are double.To prove that the kernel ofℒc is spanned by (d/dx)(ϕw(c),ψw(c)), consider the quadratic form 𝒬c(g,h) as the pairing of (g,h) against (g̃,h̃) in the Hper1([0,L])×Lper2([0,L])-Hper-1([0,L])×Lper2([0,L]) duality, where (g̃,h̃)t is the unbounded operator
(4.5)ℒ̃c:=(1-∂xx-3ϕw(c)2cc1)
applied to (g,h)t. Then ℒ̃c(g,h)t=0 implies
(4.6)-g′′+(1-c2)g-3ϕw(c)2g=0,h=-cg.
Now, from the properties of the operator ℒdn=-∂x2+w-3ϕw2 established in Theorem 4.1, it follows that g=λϕw(c)′ and h=-cg=-cλϕw(c)′=λψw(c)′, where λ≠0∈ℝ.To show that there is a single, simple, negative eigenvalue, consider𝒬c1 defined in (4.3) above. By Theorem 4.1, the operator ℒdn has exactly one negative eigenvalue which is simple, say λ0, with associated eigenfunction ζ>0. Thus, 𝒬c1 achieves a negative value and so does 𝒬c. In fact, considering ζ⃗=(ζ,-cζ), we have ℒc(ζ⃗)=Qc1(ζ)+(1/2)∥cζ-cζ∥2=Qc1(ζ)=(1/2)λ0<0. Denoting by β0 the lowest eigenvalue of ℒc, we will show that the next eigenvalue β1 is 0, which is known to be simple, and consequently β2 is in fact strictly positive. These results are proved using the (min-max) Rayley-Ritz characterization of eigenvalues (see [14, 15]), namely, (4.7)β1=max(ϕ1,ψ1)∈Xmin(g,h)∈X∖{0}〈g,ϕ1〉1+〈h,ψ1〉0=0Qc(g,h)∥g∥12+∥h∥02.
Choosing ϕ1=ζ, ψ1=0, we obtain the lower estimate (4.8)β1≥min(g,h)∈X∖{0}〈g,ζ〉1=0Qc(g,h)∥g∥12+∥h∥02.
The right-hand side of (4.8) is nonnegative on the subspace {(g,h)∈X∖{0};(g,ζ)1=0}, since Qc1(g)≥0 by Theorem 4.1. Thus, β1=0 and, from earlier considerations, β1 is simple and β2>0.The above analysis can be summarized in the form of the following theorem.Theorem 4.2.
Letℒc be the linear operator defined on  Hper2([0,L])×Hper1([0,L]) by (4.1). Then the first two eigenvalues  β0 and  β1 of  ℒc are simple and satisfy  β0<β1=0;  ζ⃗c=(ζ1,c,ζ2,c), with  ζ1,c>0 and  ϕ⃗c′ being the eigenfunctions of  β0 and  β1, respectively. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues and the mapping  c→ζ⃗c is continuous with values in  Hper2([0,L])×Hper1([0,L]).


## 5. Concavity ofd(c)

Lemma 5.1.
Letc∈(-1,1) and L>π2. Then the function d(c) is concave, provided c2<1/2 and 1-c2>2π2/L2.Remark 5.2.
Relation (1.8) implies that d′′(c)<0 is equivalent to the condition
(5.1)ddcI(ϕw(c),ψw(c))<0.Proof of Lemma5.1.
Note that(5.2)ddcI(ϕw,ψw)=ddc∫0Lϕwψw=-∫0Lϕw2dx-cddw[∫0Lϕw2dx]dwdc=-∫0Lϕw2dx+2c2ddw[∫0Lϕw2dx].
Now,
(5.3)ddw12∫0Lϕw2dx=4Lddk[K(k)E(k)]dkdw>0.
Indeed, we observe from (3.7), (3.8), and (3.11) that
(5.4)∥ϕw∥2=2η1∫0(η1/2)Ldn2(x;k)dx=8K(k)L∫0Kdn2(x;k)dx,
where we used the fact that the Jacobi elliptic function dn has fundamental period 2K and is an even function. Now, by using that ∫0Kcn2(x;k)dx=(1/k2)[E(k)-(k′)2K(k)] and dn2(x;k)=1-k2+k2cn2(x;k), it follows from (5.4) that
(5.5)12∫0Lϕw2dx=4LK(k)E(k).
Now, Proposition 3.1 and Theorem 3.2 imply that the map w→Λ(w)≡η2(w) is strictly decreasing and from (3.10), with η2=η2(w), we have that
(5.6)dkdw=12k[2η22-4wη2η2′(2w-η22)2]>0.
Thus, since k∈(0,1)→K(k)E(k) is strictly increasing (see Appendix), the claim (5.3) follows from (5.5) and (5.6).
So, from (5.2), (5.3), and (5.5), we get
(5.7)ddcI(ϕw,ψw)=-8LK(k)E(k)+16c2Lddk[K(k)E(k)]dkdw.
Now, considering the function Ψ defined by (2.12) in [12] and using (5.6), we obtain
(5.8)∂Ψ∂w=222w-η22(dK/dw)(k(η2,w))-22K(k(η2,w))(2w-η22)-(1/2)(2w-η22)=222w-η22(dK/dk)(k(η2,w))(η22/k(2w-η22)2)(2w-η22)-22K(k(η2,w))(2w-η22)-(1/2)(2w-η22),
hence
(5.9)dkdw=12k(2w-η22)2{2η22-4w[k(2w-η22)K-η22(dK/dk)k(2w-η22)K-2w(dK/dk)]}>0.
From (5.7), (5.9), and using that 2w-η22=η22/(k′)2, we obtain θ(L/8)(dI(ϕw,ψw)/dc)=θ{-EK+2c2((E2-k'2K2)/kk'2)(dk/dw)}=-θEK+2c2(E2-k'2K2)K(η22-2w)(k′)2=K{-η22E(k2η22K-2wE+2wk'2K)}+K{2c2(E2-k'2K2)(k′)2(η22-2w)}, or equivalently,
(5.10)θL8KdI(ϕw,ψw)dc=-η22E(k2η22K-2wE+2wk'2K)+2c2(k′)2(E2-k'2K2)(η22-2w)=η22(-2wk'2-η22k2)EK+(2c2(k′)2η22-4c2w(k′)2+2wη22)E2+2c2(k′)2η22K2,
where θ=η22(k2η22K-2wE+2wk'2K)<0.
Now, given that(k′)2=η22/(2w-η22), we rewrite the coefficient of E2 in (5.10) as
(5.11)2c2(k′)2η22-4c2w(k′)2+2wη22=2c2η242w-η22-4c2wη222w-η22+2wη22=2c2η24-4c2wη22+4w2η22-2wη242w-η22=η22(2w-η22)(2w-2c2)2w-η22=η22(2w-2c2).
Also, the coefficient of EK can be rewritten as η22(-2wk'2-η22k2)=2η24. Thus,
(5.12)L8KdI(ϕw,ψw)dc=-2η24EK+2η22(w-c2)E2+2c2(k′)2η22K2η22(k2η22K-2wE+2wk'2K).
We remark that we can writew as a function of complete elliptic integrals. In fact, by integrating (3.4) from 0 to L, we obtain
(5.13)w=w(c)=∫0Lϕw3(ξ)dξ∫0Lϕw(ξ)dξ,
which is well defined, since the solution ϕw is positive.
Now, using (3.7), the expression 314.01 in [11], and the fact that F(π/2;k)=K(k) (see Appendix), we obtain
(5.14)∫0Lϕw(ξ)dξ=∫0Lη1dn(η12ξ;k)dξ=2∫0η1L/2dn(y;k)dy=2∫02Kdn(y;k)dy=22∫0Kdn(y;k)dy=π2.
Similarly using (3.7), the expression 314.03 in [11], and the special values sn0=0, and snK=1, cnK=0 (see Appendix), it follows that
(5.15)∫0Lϕw3(ξ)dξ=∫0Lη13dn3(η12ξ;k)dξ=2η12∫02Kdn3(y;k)dξ=22η12∫0Kdn3(y;k)dξ=162K2L212[(1+(k′)2)π2+k2snKcnK]=4π2(1+(k′)2)K2L2.
Substituting (5.14) and (5.15) in (5.13), we deduce that
(5.16)w(c)=1-c2=4(1+(k′)2)K2L2.
Using (5.16) and η22=2w(k′)2/(1+(k′)2), the numerator of (5.12) will be positive if and only if c2(k′)2K2>(c2-w)E2+η22EK⇔(1-w)(k′)2K2>(1-2w)E2+(2w(k′)2/(1+(k′)2))EK⇔w[2E2-(k′)2K2-(2(k′)2/(1+k'2))EK]>E2-(k′)2K2⇔w>((E2-(k′)2K2)/((2E2-(k′)2K2)-(2(k′)2/(1+k'2))EK)).
Remark 5.3.
2E2-(k′)2K2-(2(k′)2/(1+k'2))EK>0 since the functions EK and E+K are strictly increasing (see Appendix).Claim 1.
(5.17)limk→0E2-(k′)2K22E2-(k′)2K2-(2(k′)2/(1+k'2))EK=25.Proof.
Indeed, denoting byf(k):=E2-(k′)2K2 and by g(k):=E2-(2(k′)2/(1+k'2))EK, we use L′Hospital’s rule to find the limit (5.17). Specifically, we show using (A.3) that
(5.18)limk→0f(j)(k)=limk→0g(j)(k)=0,(j=0,1,2,3),limk→0f(4)(k)=3π24,limk→0g(4)(k)=9π28,
which implies our claim.
Note that, bylimk→1k'2K2=0, we have that
(5.19)limk→1E2-(k′)2K2E2-(k′)2K2+E2-(2(k′)2/(1+k'2))EK=12.
Moreover, 0<((E2-(k′)2K2)/(E2-(k′)2K2+E2-(2(k′)2/(1+k'2))EK))<1/2  for allk∈(0,1), since E2-(2k'2/(1+k'2))K2>E2-k'2K2for allk∈(0,1). In, addition we get (1+k'2)K>2E, since the function m(k):=(1+k'2)K-2E has the following properties: m(0)=0 and m′(k)>0for allk∈(0,1). We conclude that the function
(5.20)f(k)f(k)+g(k)=E2-(k′)2K2E2-(k′)2K2+E2-(2(k′)2/(1+k'2))EK
is strictly positive on [0,1]. Now, continuity plus (5.17) and (5.19) implies that c0:=max0≤k≤1(f(k)/(f(k)+g(k))) satisfies 0<c0≤1/2.
This concludes the lemma.


## 6. Instability

Consider the functiond(c) defined by (1.9). We now examine the relation between concavity properties of d and the properties of the functional H near the critical point ϕ⃗c subject to the constraint I(u⃗)=I(ϕ⃗c).Theorem 6.1.
Letc≠0 be fixed. If d′′(c)<0, then there exists a curve w→Φ⃗w which satisfies I(Φ⃗w)=I(ϕ⃗c), Φ⃗c=ϕ⃗c, and on which H(u⃗) has a strict local maximum at u⃗=ϕ⃗c.Proof.
We follow the ideas of [3, 8, 9]. Let ζ⃗c be the unique, negative eigenfunction of ℒc. Define Φ⃗w:=ϕ⃗w+s(w)ζ⃗c, for w near c, where s(w) satisfies s(c)=0 and I(Φ⃗w)=I(ϕ⃗c). The function s(w) can be defined by the implicit function theorem, since
(6.1)∂∂sI(ϕ⃗w+sζ⃗c)|{s=0,w=c}=∫0L(ϕcζ2,c+ψcζ1,c)dx,
where ζ⃗c=(ζ1,c,ζ2,c) with ζ2,c=(c/(β0-1))ζ1,c, and β0 is the unique negative eigenvalue of ℒc and ψc=-cϕc, ϕc>0. Thus
(6.2)∂∂sI(ϕ⃗w+sζ⃗c)|{s=0,w=c}=-c(1+11-β0)∫0Lϕcζ1,cdx≠0.
It is easy to see that
(6.3)d2dw2I(Φ⃗w)|{w=c}=〈ℒcy⃗,y⃗〉,
where y⃗=∂Φ⃗w/∂w|{w=c}=∂ϕ⃗c/∂c+s′(c)ζ⃗c. In fact, by some calculations, we have that
(6.4)d′′(c)=-s′(c)〈I′(ϕ⃗c),ζ⃗c〉,ℒcy⃗=-I′(ϕ⃗c)+s′(c)ℒcζ⃗c,
so that
(6.5)〈ℒcy⃗,y⃗〉=-s′(c)〈I′(ϕ⃗c),ζ⃗c〉+s′(c)2〈ζ⃗c,ℒcζ⃗c〉<0
in view of the fact that d′′(c)<0.To prove the instability, we need the following lemmas which are proved in [8] and as in the analogous case of [7]; therefore, we state them without proof.Lemma 6.2.
There existϵ>0 and a unique C1  map α:Uϵ→ℝ, such that, for any u⃗=(u,v)∈Uϵ and r∈ℝ,
(6.6)〈u⃗(·+α(u⃗),∂xϕ⃗c)〉=0,α(u⃗(·+r))=α(u⃗)-r,modulotheperiod,α′(u⃗)=∂xϕ⃗c(·-α(u⃗))〈u⃗,∂x2ϕ⃗c(·-α(u⃗))〉,
where Uϵ is the “tube”
(6.7)Uϵ={u⃗∈X:infs∈ℝ∥u⃗-τs(ϕ⃗c)∥X<ϵ}.Definition 6.3.
Foru⃗∈Uϵ, define B(u⃗) by the formula
(6.8)B(u⃗)=y⃗(·-α(u⃗))-K∂xα′(u⃗),
where K=(0110).By Lemma6.2, B may also be expressed as
(6.9)B(u⃗)=y⃗(·-α(u⃗))-〈Ky⃗(·-α(u⃗)),u⃗〉〈∂x2ϕ⃗c(·-α(u⃗)),u⃗〉K∂x2ϕ⃗c(·-α(u⃗)).Lemma 6.4.
B is a C1 function from Uϵ into X, B commutes with translations, B(ϕ⃗c)=y⃗, and 〈B(u⃗),Ku⃗〉=0, for any u⃗∈Uϵ.Lemma 6.5.
There exists aC1 function(6.10)Π:{v⃗∈Uϵ:I(v⃗)=I(ϕ⃗c)}→ℝ
which is invariant under translations, such that if v⃗∈Uϵ with I(v⃗)=I(ϕ⃗c) and v⃗ not being a translation of ϕ⃗c, we have
(6.11)H(ϕ⃗c)<H(v⃗)+Π(v⃗)〈H′(v⃗),B(v⃗)〉.Lemma 6.6.
The curvew→Φ⃗w constructed in Theorem 6.1 satisfies H(Φ⃗w)<H(ϕ⃗c) for w≠c, and 〈H′(Φ⃗w),B(Φ⃗w)〉 changes sign as w passes through c, with c≠0.


### 6.1. Proof of Theorem1.2

First, considerc≠0 with |c|<1. Let ϵ>0 be given small enough. By Lemma 6.6, we can choose u⃗0∈X arbitrarily close to ϕ⃗c such that I(u⃗0)=I(ϕ⃗c), H(u⃗0)<H(ϕ⃗c) e 〈H′(u⃗0),andB(u⃗0)〉>0. To prove the instability of ϕ⃗c, it suffices to show that there are some elements u⃗0∈X which are close to ϕ⃗c but for which the solution u⃗ of (1.2) with initial data u⃗0 exits from Uϵ in finite time. Let [0,t1) denote the maximal interval for which u⃗(·,t) lies continuously in Uϵ. By Lemma 2.1, t1>0. Let T be the maximum existence time of solution u⃗ in (1.1) with initial data u⃗0. If T is finite, then we have the X-instability for ϕ⃗c by definition. So we may assume that T=+∞ and it suffices to show that t1<+∞.In view of Lemma2.1 and Proposition 2.2, u⃗ has the following properties: u⃗∈C([0,t1];X),u⃗(0,x)=u⃗0(x), and I(u⃗(t))andH(u⃗(t)) are constant for t∈[0,t1). Let ρ(t)=α(u⃗(t)), where α is defined by Lemma 6.2 and define (6.12)y⃗(z)=dΦ⃗cdc=dϕ⃗cdc+s′(c)χ⃗c,Y⃗(x)=∫0xKy⃗(z)dz,A(t)=∫0LY⃗(x-ρ(t))·u⃗(x,t)dx,for0≤t<t1,γ⃗=∫0LKy⃗(x)dx.
The function A(t) serves as a Lyapunov function in our argument.Now we estimatedA/dt. By differentiation, (6.13)dAdt=-ρ′(t)∫0LKy⃗(x-ρ(t))·u⃗(x,t)dx+∫0LY⃗(x-ρ(t))·∂u⃗∂t(x,t)dx,
where ∂u⃗/∂t is in distribution sense. Since ρ′(t)=dα(u⃗(t))/dt=〈α′(u⃗),(∂u⃗/∂t)〉, (6.14)dAdt=〈-〈Ky⃗(·-ρ),u⃗〉α′(u⃗)+Y⃗(·-ρ),∂u⃗∂t〉.Since(6.15)du⃗dt=JH′(u⃗),whereJ=∂xK,∂xY⃗=Ky⃗,
it follows that (6.16)dAdt=〈-〈Ky⃗(·-ρ(t)),u⃗〉∂xα′(u⃗)-Ky⃗(·-ρ(t)),KH′(u⃗)〉=〈-〈Ky⃗(·-ρ(t)),u⃗〉K∂xα′(u⃗)-y⃗(·-ρ(t)),H′(u⃗)〉=-〈B(u⃗),H′(u⃗)〉.
Since (6.17)0<H(ϕ⃗c)-H(u⃗0)=H(ϕ⃗c)-H(u⃗(t)),
Lemma 6.5 implies that (6.18)0<Π(u⃗(t))〈B(u⃗(t)),H′(u⃗(t))〉.
By 〈B(u⃗0),H′(u⃗0)〉>0 and continuity, we obtain 〈B(u⃗(t)),H′(u⃗(t))〉>0 for 0<t<t1. Moreover, since u⃗(t)∈Uϵ and Π(ϕ⃗c)=0, we may assume that 0<Π(u⃗(t))<1 for 0<t<t1, by choosing ϵ smaller if necessary. Therefore, for all t∈[0,t1), by Lemma 6.5, (6.19)〈B(u⃗(t)),H′(u⃗(t))〉≥Π(u⃗(t))〈B(u⃗(t)),H′(u⃗(t))〉>H(ϕ⃗c)-H(u⃗(t))=H(ϕ⃗c)-H(u⃗0)>0.
Hence, (6.20)-dAdt≥H(ϕ⃗c)-H(u⃗0)>0.
So we conclude that t1<+∞.Finally, consider the casec=0 and d′′(0)<0.LetN={c∣thednoidalwaveϕ⃗cisX-unstable}. The curve c→ϕ⃗c is continuous in X. From the definition of stability, the set N is closed. Since (6.21)d′′(c)=〈I′(ϕ⃗c),∂ϕ⃗c∂c〉
by Theorem 3.2, d′′(c) is continuous on (-1-(2π2/L2),1-(2π2/L2))⊂(-1,1). For any nonzero sequence cn→0, we have d′′(cn)→d′′(0)<0. Hence d′′(cn)<0 for large n. Thus cn∈N, for large n. It follows that c=0∈N.


## 6.1. Proof of Theorem1.2

First, considerc≠0 with |c|<1. Let ϵ>0 be given small enough. By Lemma 6.6, we can choose u⃗0∈X arbitrarily close to ϕ⃗c such that I(u⃗0)=I(ϕ⃗c), H(u⃗0)<H(ϕ⃗c) e 〈H′(u⃗0),andB(u⃗0)〉>0. To prove the instability of ϕ⃗c, it suffices to show that there are some elements u⃗0∈X which are close to ϕ⃗c but for which the solution u⃗ of (1.2) with initial data u⃗0 exits from Uϵ in finite time. Let [0,t1) denote the maximal interval for which u⃗(·,t) lies continuously in Uϵ. By Lemma 2.1, t1>0. Let T be the maximum existence time of solution u⃗ in (1.1) with initial data u⃗0. If T is finite, then we have the X-instability for ϕ⃗c by definition. So we may assume that T=+∞ and it suffices to show that t1<+∞.In view of Lemma2.1 and Proposition 2.2, u⃗ has the following properties: u⃗∈C([0,t1];X),u⃗(0,x)=u⃗0(x), and I(u⃗(t))andH(u⃗(t)) are constant for t∈[0,t1). Let ρ(t)=α(u⃗(t)), where α is defined by Lemma 6.2 and define (6.12)y⃗(z)=dΦ⃗cdc=dϕ⃗cdc+s′(c)χ⃗c,Y⃗(x)=∫0xKy⃗(z)dz,A(t)=∫0LY⃗(x-ρ(t))·u⃗(x,t)dx,for0≤t<t1,γ⃗=∫0LKy⃗(x)dx.
The function A(t) serves as a Lyapunov function in our argument.Now we estimatedA/dt. By differentiation, (6.13)dAdt=-ρ′(t)∫0LKy⃗(x-ρ(t))·u⃗(x,t)dx+∫0LY⃗(x-ρ(t))·∂u⃗∂t(x,t)dx,
where ∂u⃗/∂t is in distribution sense. Since ρ′(t)=dα(u⃗(t))/dt=〈α′(u⃗),(∂u⃗/∂t)〉, (6.14)dAdt=〈-〈Ky⃗(·-ρ),u⃗〉α′(u⃗)+Y⃗(·-ρ),∂u⃗∂t〉.Since(6.15)du⃗dt=JH′(u⃗),whereJ=∂xK,∂xY⃗=Ky⃗,
it follows that (6.16)dAdt=〈-〈Ky⃗(·-ρ(t)),u⃗〉∂xα′(u⃗)-Ky⃗(·-ρ(t)),KH′(u⃗)〉=〈-〈Ky⃗(·-ρ(t)),u⃗〉K∂xα′(u⃗)-y⃗(·-ρ(t)),H′(u⃗)〉=-〈B(u⃗),H′(u⃗)〉.
Since (6.17)0<H(ϕ⃗c)-H(u⃗0)=H(ϕ⃗c)-H(u⃗(t)),
Lemma 6.5 implies that (6.18)0<Π(u⃗(t))〈B(u⃗(t)),H′(u⃗(t))〉.
By 〈B(u⃗0),H′(u⃗0)〉>0 and continuity, we obtain 〈B(u⃗(t)),H′(u⃗(t))〉>0 for 0<t<t1. Moreover, since u⃗(t)∈Uϵ and Π(ϕ⃗c)=0, we may assume that 0<Π(u⃗(t))<1 for 0<t<t1, by choosing ϵ smaller if necessary. Therefore, for all t∈[0,t1), by Lemma 6.5, (6.19)〈B(u⃗(t)),H′(u⃗(t))〉≥Π(u⃗(t))〈B(u⃗(t)),H′(u⃗(t))〉>H(ϕ⃗c)-H(u⃗(t))=H(ϕ⃗c)-H(u⃗0)>0.
Hence, (6.20)-dAdt≥H(ϕ⃗c)-H(u⃗0)>0.
So we conclude that t1<+∞.Finally, consider the casec=0 and d′′(0)<0.LetN={c∣thednoidalwaveϕ⃗cisX-unstable}. The curve c→ϕ⃗c is continuous in X. From the definition of stability, the set N is closed. Since (6.21)d′′(c)=〈I′(ϕ⃗c),∂ϕ⃗c∂c〉
by Theorem 3.2, d′′(c) is continuous on (-1-(2π2/L2),1-(2π2/L2))⊂(-1,1). For any nonzero sequence cn→0, we have d′′(cn)→d′′(0)<0. Hence d′′(cn)<0 for large n. Thus cn∈N, for large n. It follows that c=0∈N.



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*Source: 637497-2010-07-12.xml*