Volume 2010, Article ID 637497,21pages doi:10.1155/2010/637497

*Research Article*

**On the Instability of a Class of**

**Periodic Travelling Wave Solutions of** **the Modified Boussinesq Equation**

**Lynnyngs Kelly Arruda**

*Departamento de Matem´atica, Universidade Federal de S˜ao Carlos, Caixa Postal 676, S˜ao Carlos CEP,*
*13565-905 S˜ao Paulo, Brazil*

Correspondence should be addressed to Lynnyngs Kelly Arruda,lynnyngs@dm.ufscar.br Received 1 February 2010; Accepted 7 May 2010

Academic Editor: Prabir Daripa

Copyrightq2010 Lynnyngs Kelly Arruda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with instability of periodic travelling wave solutions of the modified
*Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental period L 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.

**1. Introduction**

The original Boussinesq equations are among the classical models for the propagation of
small amplitude, planar long waves on the surface of water1,2. 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 3below. 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
solutions*φx*−*ct*of the modified Boussinesq equation

*u**tt*−*u**xx*

*u*^{3}*u**xx*

*xx*0. 1.1

The above equation1.1, has the following equivalent form as a Hamiltonian system
*u*_{t}*v*_{x}*,*

*v**t*

*u*−*u**xx*−*u*^{3}

*x*

1.2

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

*Hu, v * 1
2

_{L}

0

*u*^{2}*v*^{2}*u*^{2}* _{x}*−

*u*

^{4}2

*dx* 1.3

does not depend on the time*t. Another conservation law is the momentum*

*Iu, v *
_{L}

0

*uvdx* 1.4

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, let*τ**s*be the translation by*s,τ**s**φx *
*φx*sfor*x*∈Rand let→−

*φ** _{c}* φ

*c*x−

*ct, ψ*

*c*x−ctbe an

*L-periodic travelling wave solution*to system1.2, where

*φ*

*c*:R → R,

*ψ*

*c*:R → R,

*L >*0 is the period of

*φ*

*c*and

*ψ*

*c*, and

*c*is the wave’s speed of propagation. If we define the→−

*φ** _{c}*-orbit to be the setΩ

^{→}

^{−}

_{φ}*c* {→−

*φ** _{c}*·

*s, s*∈R},

−

→*φ** _{c}*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. Let*X*be a Hilbert space.

*Definition 1.1* Orbital Stability. Let→−

*φ** _{c}* φ

*c*x −

*ct, ψ*

*c*x−

*ct*∈

*X*be an

*L-periodic*travelling wave solution to system1.2. We say that the orbit Ω

^{→}

^{−}

_{φ}*c* is stable in the*X-sense*
by the flow of system1.2if for each* >*0 there exists*δ* *δ>*0 such that if→−*u*_{0} ∈*X*and
inf* _{s∈R}*−→

*u*0−

*τ*

*s*→−

*φ*_{c}_{X}*< δ,*then the solution→−*ut*of1.2with→−*u0 *→−*u*0satisfies, for all*t*for
which→−*u* u, vexists,

inf*s∈R*

→−*ut*−*τ**s*

→−
*φ*_{c}

*X* *< .* 1.5

Otherwise, we say thatΩ^{→}^{−}_{φ}

*c* is*X-unstable.*

Here,*X* : *H*_{per}^{1} 0, L×*L*^{2}_{per}0, L.The choice of norm in1.5is dictated by the
form of the Hessian or “linearized Hamiltonian”*H*^{}→−

*φ*_{c}*cI*^{}→−

*φ** _{c}*and varies from problem
to problem.

Inserting the*L-periodic travelling wave solution*→−

*φ** _{c}* φ

*c*x−

*ct, ψ*

*c*x−

*ct*in1.2 leads to the system

−cφ_{c}^{}ξ *ψ*_{c}^{}ξ,

−cψ_{c}^{}ξ

*φ** _{c}*−

*φ*

^{}

*−*

_{c}*φ*

_{c}^{3}

_{}

ξ, 1.6

where ’ connotes*d/dξ*and*ξ* *x*−*ct. Integrating the latter system, we obtain the nonlinear*
system

−cφξ *ψξ K*_{1}*,*

−cψξ *φξ*−*φ*^{}ξ−*φ*^{3}ξ *K*2*,*

1.7

where*K*_{1}*, K*_{2}are integration constants, which will be considered equal to zero here. Then, we
obtain

*H*^{}*cI*^{} −→
*φ*_{c}

0. 1.8

Next observe that relation1.8characterizes→−

*φ** _{c}* φ

*c*

*, ψ*

*c*as a critical point of

*H*subject to the constraint

*Iu, v*

*I*φ

*c*

*, ψ*

*c*.In order to prove instability for→−

*φ** _{c}*, we will examine the
relation between the concavity properties of the function

*dc H*→−
*φ** _{c}*·

*cI*→−
*φ** _{c}*·

*,* 1.9

and the properties of the functional *H* near the critical point→−

*φ** _{c}* under the constraint

*I*constant.

Bona and Sachs in3proved that the well-known solitary waves→−

*φ* φ*c*x−ct, ψ*c*x−

*ct*of the generalized Boussinesq equation
*u*_{t}*v*_{x}*,*

*v**t* *u*−*u**xx*−*u*^{p}* _{x}* 1.10

are stable in the*H*^{1}R×*L*^{2}Rnorm for speeds*c*such thatp−1/4*< c*^{2}*<*1 if*d*in1.9is 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*dc*is concave. The proof follows the main ideas of Liu4 see also
Bona et al. in7. Diﬀerently from the solitary wave solutions case, we do not know explicit
periodic travelling wave solutions in the*x-variable for the system*1.10for 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 paper6. Regarding the classical case*p*2, in5the
author proves nonlinear stability properties of a class of*L-periodic travelling wave solutions,*
*called cnoidal waves, in the energy spaceH*_{per}^{1} 0, L×*L*^{2}_{per}0, L, by periodic disturbances
with period*L.*

In this paper, we first show the existence of a smooth curve *c* → →−

*φ** _{c}* φ

*c*

*, ψ*

*of dnoidal wave solutions to system1.2, with a fixed period*

_{c}*L*Theorem 3.2below. 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→−

*φ** _{c}*, given by
Theorem 3.2below, is the following.

**Theorem 1.2**Instability Theorem. Let*c* ∈ −1,1*and* *L > π*√

*2. Then the orbit*Ω^{→}^{−}_{φ}

*c* *is* *X-*

*unstable with respect to the flow of the modified Boussinesq equation, providedc*^{2} *<*1/2 and 1−*c*^{2} *>*

2π^{2}*/L*^{2}*.*

The plan of this paper is as follows. A discussion of the evolution equation1.1and its natural invariants is given inSection 2. InSection 3we introduce a smooth family{→−

*φ** _{c}*}

*on the parameter*

_{c}*c, of positive dnoidal wave solutions to system*1.2, with a fixed period

*L*Theorem 3.2belowand inSection 4we present a complete study of the spectrum of the operatorL

*c*. The existence of the smooth curve

*c*→→−

*φ** _{c}* will allow us to diﬀerentiate the
function

*dc. Then, in section 5, we prove thatdc*is indeed concave, for a certain range of speeds and periods of→−

*φ** _{c}*, which will imply our result. InSection 6the Lyapunov functional
8,9is 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 of→−

*φ* is established with respect to perturbations of
periodic functions of the same period*L*in*X.*

The following notation will be used:

*f, g*

0

*f, g*

*L*^{2}_{per0,L}
_{L}

0

*fgdx,*
*f, g*

1

*f, g*

*H*_{per0,L}^{1}
_{L}

0

*fgdx*
_{L}

0

*f*^{}*g*^{}*dx,*
*f*

0*f*

*L*^{2}_{per0,L}
_{L}

0

*f*^{2}*dx*
_{1/2}

*,*
*f*

1*f*

*H*_{per0,L}^{1}
_{L}

0

*f*^{2}*dx*
_{L}

0

*f*^{}^{2}*dx*
1/2

*,*
*f, g* *,*u, v

*f, g* *,*u, v

*L*^{2}_{per0,L}×L^{2}_{per0,L}
_{L}

0

*fudx*
_{L}

0

*gvdx,*
*f, g* *f, g* * _{L}*2

per0,L×L^{2}_{per0,L}
_{L}

0

*f*^{2}*dx*
_{L}

0

*g*^{2}*dx*
1/2

*,*
*f, g* _{X}*f, g* * _{H}*1

per0,L×L^{2}_{per0,L}
_{L}

0

*f*^{2}*dx*
_{L}

0

*f*^{}^{2}*dx*
_{L}

0

*g*^{2}*dx*
_{1/2}

*.*

1.11

**2. The Evolution Equation**

The next lemma is the periodic version of a particular case of4, Lemma 1.1.

* Lemma 2.1. Let*→−

*u*0 u0

*, v*0 ∈

*X*≡

*H*

_{per}

^{1}0, L×

*L*

^{2}

_{per}0, L. Then there exist

*T >*

*0 and a*

*uniquely weak solution*→−

*u*u, v

*of*1.2

*with*→−

*u0*→−

*u*

_{0}

*.*

*Proof. In order to obtain the existence of weak solutions for the system*1.2, we consider the
approximate problem

−

→*u**t**A*→−*uF*→−*u* *,*

−

→*u0 *→−*u*^{n}_{0}*,* 2.1

with→−*u*^{n}_{0} ∈*DA H*_{per}^{3} 0, L×*H*_{per}^{1} 0, Land→−*u*^{n}_{0} → −→*u*0in*X, where*

*A*

0 −∂*x*

−∂*x**∂*^{3}* _{x}* 0

2.2

and −A is the infinitesimal generator of a *C*^{0} group of unitary operators in *X* and *F*
*Ft, u, v *

−∂*x*0u^{3}

. Since*F* ∈ *C*^{∞}, the map u, v → 0, ∂*x*u^{3} is locally Lipschitz on
*X. But then for all*→−*u*^{n}_{0} ∈*DA, there exists aT**n* *>*0 such that the initial value problem2.1
has a unique solution→−*u** ^{n}*∈

*C0, T*

*n*;

*DA*∩

*C*

^{1}0, T

*n*;

*X. Moreover, ifT*

_{n}*<*∞, then

*t→*lim*T**n*

→−*u** ^{n}*t

*X* ∞, 2.3

by the semigroup theory10. By2.1, we estimate on0, T*n*
1

2
*d*
*dt*

→−*u** ^{n}*t

^{2}

*X* →−*u** ^{n}*t, ∂

*t*→−

*u*

*t*

^{n}*X* →−*u*^{n}*,*−A→−*u*^{n}*F*→−*u*^{n}

*X*

≤

_{L}

0

*∂** _{x}*u

^{n}^{3}t·

*v*

*tdx*

^{n}≤∂*x*u^{n}^{3}t

0v* ^{n}*t

_{0}3

_{L}

0

u* ^{n}*t

^{2}u

*t*

^{n}

_{x}^{2}

*dx*

_{1/2}

v* ^{n}*t

_{0}

≤3u* ^{n}*t

^{2}

_{∞}

_{L}0

u^{n}* _{x}*t

^{2}

*dx*

_{1/2}

v* ^{n}*t

_{0}

≤3SLu* ^{n}*t

^{2}

_{1}u

^{n}

_{x}_{0}v

*t*

^{n}_{0}

≤3SL→−*u** ^{n}*t

^{2}

*X*

→−*u*^{n}^{2}

*X**,*

2.4

where we used in the first equality above that−A→−*u** ^{n}*∈

*X.*

Consider now*fs * 3SLs, which is a continuous, positive and increasing function
onR^{}. Then by Gronwall’s inequality, it follows that

→−*u** ^{n}*t

^{2}

*X*≤→−*u*^{n}_{0}^{2}

*X*exp
*t*

0

*f*

→−*u** ^{n}*τ

^{2}

*X*

*dτ*

*,* on0, T*n*. 2.5

We compare−→*u** ^{n}*t

^{2}

*with the maximal solution*

_{X}*yt*

*yt*≡ 1

sup* _{n}*→−

*u*

^{n}_{0}

*X*

_{2}

1−3St

sup* _{n}*→−

*u*

^{n}_{0}

*X*

2*,*

*t*∈0, T0≡

⎡

⎢⎣0, 1 3S

sup* _{n}*→−

*u*

^{n}_{0}

*X*

_{2}

⎞

⎟⎠

2.6

of the scalar Cauchy problem

*dy*
*dt* *f*

*y* *y,*
*y0 y*_{0}sup

*n*

→−*u*^{n}_{0}^{2}

*X**.*

2.7

It follows that

→−*u** ^{n}*t

^{2}

*X*≤*yt,* on0, T*n*∩0, T0. 2.8

Let*T < T*_{0}. Then→−*u** ^{n}*is defined on0, Tfor all

*n. Moreover,*→−

*u*

*t*

^{n}^{2}

*X* ≤*C*_{0}*y*_{0} *K*^{2} 2.9

on0, T, where*K*is a constant independent of*n, since by*2.5,2.8, and the fact that*yt*
is bounded on0, T, we have the following inequality on0, T:

→−*u** ^{n}*t

^{2}

*X* ≤→−*u*^{n}_{0}^{2}

*X*exp
_{t}

0

*f*

→−*u** ^{n}*τ

^{2}

*X*

*dτ*

≤→−*u*^{n}_{0}^{2}

*X*exp
_{T}

0

*f*

*yτ* *dτ*

≤*C*0T→−*u*^{n}_{0}^{2}

*X* ≤*C*0Ty0*.*

2.10

Finally, from2.9 and standard weak limit arguments, we have the existence of a
unique solution→−*ut*∈*C0, T*;*X.*

* Proposition 2.2. The unique solution*→−

*utof*1.2

*with initial data*→−

*u*0 →−

*u*

_{0}

*, which is given by*

*Lemma 2.1, satisfies*

*H*→−*ut* *Hu, v constant,* 0,

*I*→−*ut* *Iu, v constant,* 0. 2.11

The proof is elementary.

**3. Existence of a Smooth Curve of Dnoidal Wave Solutions with** **a Fixed Period** *L* **for the System** 1.2

This section is devoted to establish the existence of a smooth curve of periodic travelling wave solutions for the system1.2, which are solutions of the form

−

→*ux, t ux, t, vx, t *

*φx*−*ct, ψx*−*ct* *.* 3.1

Substituting3.1in1.2leads to the system

−cφ^{}ξ *ψ*^{}ξ,

−cψ^{}ξ

*φ*−*φ*^{}−*φ*^{3}_{}

ξ, 3.2

where ’ denotes*d/dξ*and*ξx*−*ct. Integrating*3.2, we obtain the nonlinear system

−cφξ *ψξ K*_{1}*,*

−cψξ *φξ*−*φ*^{}ξ−*φ*^{3}ξ *K*_{2}*,* 3.3

where*K*_{1}*, K*_{2}are integration constants, which will be considered equal to zero here. Then,*φ*
must satisfy

*φ*^{}−*wφφ*^{3}0, 3.4

where*wwc *1−*c*^{2}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 parameterc. In order to do this, we first observe*
from3.4that*φ*satisfies the first-order equation

*φ*^{} ^{2} 1
2

−φ^{4}2wφ^{2}4B*φ*

1 2

*η*^{2}_{1}−*φ*^{2}

*φ*^{2}−*η*_{2}^{2}

*,* 3.5

where*B**φ* is an integration constant and−η1,*η*1,−η2,*η*2are the real zeros of the polynomial
*p** _{φ}*t −t

^{4}2wt

^{2}4B

*, which satisfy the relations*

_{φ}2w*η*^{2}_{1}*η*^{2}_{2}*,*

4B* _{φ}* −η

_{1}

^{2}

*η*

^{2}

_{2}

*.*3.6

Moreover, we assume without lost of generality that*η*_{1} *> η*_{2} *>* 0 and we obtain from3.5
that*η*_{2}≤*φ*≤*η*_{1}. By defining*ϕφ/η*_{1}and*k*^{2} η^{2}_{1}−*η*^{2}_{2}/η^{2}_{1},3.5becomesϕ^{}^{2} η^{2}_{1}*/21*−
*ϕ*^{2}ϕ^{2}−1*k*^{2}. We also impose the crest of the wave to be at*ξ* 0, that is,*φ0 *1. Now,
we define a further variable*ψ* via the relation*ϕ*^{2} 1−*k*^{2}sin^{2}*ψ* and so we get thatψ^{}^{2}
η_{1}^{2}*/21*−*k*^{2}sin^{2}*ψ.*Then we obtain for*lη*_{1}*/*√

2 that_{ψξ}

0 dt/

1−*k*^{2}sin^{2}*t lξ.*Therefore,

from the definition of the Jacobian elliptic function*y* snu;*k see in the appendix or in*
Byrd and Friedman11, we can write the last equality as sin*ψ*snlξ;*k*and hence*ϕξ *
1−*k*^{2}sn^{2}lξ;*k * dnlξ;*k.Returning to the initial variable, we obtain the called dnoidal*
*wave solution associated to*3.4,

*φξ*≡*φ*

*ξ;η*_{1}*, η*_{2} *η*_{1}dn
*η*_{1}

√2*ξ;k*

3.7

with

*k*^{2} *η*^{2}_{1}−*η*^{2}_{2}

*η*^{2}_{1} *,* *η*_{1}^{2}*η*^{2}_{2}2w, 0*< η*2*< η*1*.* 3.8

Next, dn has fundamental period 2K, dnu2K;*k * dnu;*k, whereK* *Kk*represents
the complete elliptic integral of the first kindsee appendix; it follows that the dnoidal wave
*φ*in3.7has fundamental period,*T** _{φ}*, given by

*T** _{φ}*≡ 2√
2

*η*_{1} *Kk.* 3.9

Now, we show that*T**φ* *>* √
2π/√

*w. First, we expressT**φ* as a function of*η*2 and*w. In fact,*
for every*η*_{2} ∈0,√

*w, there is a uniqueη*_{1} ∈√
*w,*√

2wsatisfying the first relation in3.6,
namely,*η*_{1}

2w−*η*^{2}_{2}. So, from3.9we obtain

*T**φ*

*η*2*, w* 2√
2

2w−*η*^{2}_{2}
*K*

*k*
*η*2

*,* with*k*^{2}

*η*2*, w* 2w−2η_{2}^{2}

2w−*η*^{2}_{2} *.* 3.10

Then, by fixing*w >*0, we have that*T** _{φ}* → ∞as

*η*

_{2}→ 0 and

*T*

*η2 → π√ 2/√*

_{φ}*w*
as *η*2 → √

*w. So, since the mapping* *η*2 → *T**φ**w*η2 is strictly decreasing see proof of
Proposition 3.1, it follows that*T*_{φ}*>*√

2π/√
*w.*

Now, we obtain a dnoidal wave solution with period*L. Forw*0 *>* 2π^{2}*/L*^{2}, there is a
unique*η*2,0 ∈ 0,√

*w*0such that*T**φ*η2,0*, w*0 *L. So, forη*1,0such that*η*_{1,0}^{2} *η*^{2}_{2,0} 2w0, the
dnoidal wave*φ· φ·, η*1,0*, η*_{2,0}has a fundamental period*L*and satisfies3.4with*ww*_{0}.
By the above analysis the dnoidal wave*φ·, η*1*, η*2in3.7is completely determined
by*w*and*η*2and will be denoted by*φ**w*·;*η*2or*φ**w*.

The next result, which corresponds to Theorem 2.1 and Corollary 2.2 in 12, is concerned with the existence of a smooth curve of dnoidal wave solutions for3.4.

**Proposition 3.1. Let**L >*0 be arbitrary but fixed. Considerw*0 *>* 2π^{2}*/L*^{2} *and the uniqueη*2,0
*η*_{2}w0∈0,√

*w*_{0}*such thatT*_{φ}_{w}

0 *L. Then,*

1*there exist an interval* Iw0 *aroundw*0*, an interval* *Jη*2,0*around* *η*2,0*, and a unique*
*smooth function*Λ:Iw0 → *Jη*2,0*such that*Λw0 *η*_{2,0}*and*

2√ 2

2w−*η*^{2}_{2}

*Kk L,* 3.11

*wherew*∈ Iw0,*η*2 Λw, and*k*^{2}*k*^{2}w∈0,1*is defined by*3.10;

2*the positive dnoidal wave solution in* 3.7, *φ** _{w}*·;

*η*

_{1}

*, η*

_{2}, determined by

*η*

_{1}≡

*η*

_{1}w 2w−

*η*

^{2}

_{2}

*,*

*η*2 ≡

*η*2w, has fundamental period

*L*

*and satisfies*3.4. Moreover, the

*mappingw*∈ Iw0 →

*φ*

*∈*

_{w}*H*

_{per}

^{1}0, Lis a smooth function;

3Iw0*can be chosen as*2π^{2}*/L*^{2}*,*∞;

4*the mapping*Λ:2π^{2}*/L*^{2}*,*∞ → *Jη*2,0*is strictly decreasing.*

*Proof (see [12]). From this result we conclude the following existence theorem.*

**Theorem 3.2. Let***L > π*√

*2. Then there exists a smooth curve of dnoidal wave solutions for the*
*system*1.2*inH*_{per}* ^{n}* 0, L×

*H*

_{per}

*0, L,*

^{m}*n, m*≥

*0 which satisfy the system*3.3

*with integration*

*constantsK*

_{1}

*K*

_{2}

*0; this curve is given, forwc*1−

*c*

^{2}

*, by*

*c*∈

⎛

⎝−

!
1−2π^{2}

*L*^{2} *,*

!
1− 2π^{2}

*L*^{2}

⎞

⎠−→

*φ*_{wc}*, ψ*_{wc}*.* 3.12

*Moreover,φ**c*ξ:*φ** _{wc}*ξ

2w−*η*^{2}_{2}dn

2w−*η*_{2}^{2}*/*√

2ξ;*k, ψ**wc* −cφ*wc**,where*
*the smooth functionη*_{2}≡*η*_{2}wc*is given byProposition 3.1andkkwcby*3.10.

*Remark 3.3.* *∂*→−

*φ*_{c}*/∂c*is in*H*_{per}^{∞}0, L×*H*_{per}^{∞}0, Las soon as in*H*_{per}^{1} 0, L×*H*_{per}^{1} 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
L*c**H*^{}

*φ*_{wc}*, ψ*_{wc}*cI*^{}

*φ*_{wc}*, ψ** _{wc}* 4.1

determined by the periodic solutionsφ*wc**, ψ** _{wc}* found in Theorem 3.2. We compute the
Hessian operatorL

*c*by calculating the associated quadratic form, which is denoted byQ

*c*. By definition,Q

*c*g, his the coeﬃcient of

^{2}in

*H*

*φ*_{wc}*g, ψ*_{wc}*h* *cI*

*φ*_{wc}*g, ψ*_{wc}*h* *,* 4.2

and so is given by

Q*c*

*g, h*
_{L}

0

"

1 2

*g*^{2}*g*_{x}^{2}*h*^{2}

−3

2*φ*^{2}_{wc}*g*^{2}*cgh*

#
*dx*

_{L}

0

"

1 2

1−*c*^{2}

*g*^{2}*g*^{}^{2} −3φ^{2}_{wc}*g*^{2}
1

2

*hcg* ^{2}*dx*

#

:Q^{1}_{c}*g* 1

2*hcg*^{2}

0*.*

4.3

Note thatQ*c*is the sum of the quadratic formQ^{1}* _{c}*associated to the operator−d

^{2}

*/dx*

^{2}1−

*c*

^{2}− 3φ

^{2}

*and the nonnegative term1/2h*

_{w}*cg*

^{2}

_{0}. From3.2for the dnoidal waveφ

*wc*

*, ψ*

*, it follows that*

_{wc}*g*

*φ*

^{}

*and*

_{wc}*h*

*ψ*

_{wc}^{}satisfy L

*c*g, h 0. To see that this is the only eigenfunction corresponding to the eigenvalue zero and the other expected properties of the operatorL

*c*, we will first consider the following result about the periodic eigenvalue problem:

Ldn*ξ*:

− *d*^{2}

*dx*^{2} *w*−3φ^{2}_{w}

*ξλ,*
*ξ0 ξL, ξ*^{}0 *ξ*^{}L,

4.4

where*φ** _{w}*is given byProposition 3.1.

The following result is a consequence of the Floquet theoryMagnus and Winkler13 and can be found in12.

* Theorem 4.1. Let*Ldn

*be the linear operator defined onH*

_{per}

^{2}0, L

*by*4.4. Then the first three

*eigenvaluesβ*1

*,β*2

*, andβ*3

*of*Ldn

*are simple, and satisfyβ*1

*<*0

*β*2

*< β*3

*, andφ*

^{}

*is the eigenfunction*

*ofβ*

_{2}

*. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues which are double.*

To prove that the kernel of L*c* is spanned by d/dxφ*wc**, ψ** _{wc}*, consider the
quadratic formQ

*c*g, has the pairing ofg, hagainst

*g,h*in the

*H*

_{per}

^{1}0, L×L

^{2}

_{per}0, L−

*H*

_{per}

^{−1}0, L×

*L*

^{2}

_{per}0, Lduality, where

*g,h*

*is the unbounded operator*

^{t}L*c*:

1−*∂**xx*−3φ^{2}_{wc}*c*

*c* 1

4.5

applied tog, h* ^{t}*. ThenL

*c*g, h

*0 implies*

^{t}−g^{}
1−*c*^{2}

*g*−3φ_{wc}^{2} *g* 0,
*h*−cg.

4.6

Now, from the properties of the operatorLdn −∂^{2}_{x}*w*−3φ_{w}^{2} established inTheorem 4.1, it
follows that*gλφ*_{wc}^{} and*h*−cg −cλφ^{}_{wc}*λψ*_{wc}^{} , where*λ /*0∈R.

To show that there is a single, simple, negative eigenvalue, considerQ^{1}* _{c}* defined in
4.3above. ByTheorem 4.1, the operatorLdn has exactly one negative eigenvalue which is
simple, say

*λ*

_{0}, with associated eigenfunction

*ζ >*0. Thus,Q

^{1}

*achieves a negative value and so doesQ*

_{c}*c*. In fact, considering→−

*ζ* ζ,−cζ, we haveL*c*→−

*ζ* *Q*^{1}* _{c}*ζ 1/2cζ−

*cζ*

^{2}

*Q*

^{1}

*ζ 1/2λ0*

_{c}*<*0.Denoting by

*β*

_{0}the lowest eigenvalue ofL

*c*, we will show that the next eigenvalue

*β*1is 0, which is known to be simple, and consequently

*β*2is in fact strictly positive.

These results are proved using the min-max Rayley-Ritz characterization of eigenvalues see14,15, namely,

*β*1 max

φ1*,ψ*1∈X min
^{g,h}^{∈X\{0}}

^{g,φ}^{1}1^{h,ψ}^{1}00

*Q*_{c}*g*^{2}*g, h*

1h^{2}_{0}*.* 4.7

Choosing*φ*1*ζ,ψ*10, we obtain the lower estimate

*β*_{1} ≥ min

g,h∈X\{0}

* ^{g,ζ}*10

*Q**c*

*g, h*
*g*^{2}

1h^{2}_{0}*.* 4.8

The right-hand side of4.8is nonnegative on the subspace{g, h ∈ *X*\ {0};g, ζ_{1} 0},
since*Q*^{1}* _{c}*g ≥ 0 byTheorem 4.1. Thus,

*β*1 0 and, from earlier considerations,

*β*1 is simple and

*β*2

*>*0.

The above analysis can be summarized in the form of the following theorem.

* Theorem 4.2. Let*L

*c*

*be the linear operator defined onH*

_{per}

^{2}0, L×

*H*

_{per}

^{1}0, L

*by*4.1. Then the

*first two eigenvaluesβ*0

*andβ*1

*of*L

*c*

*are simple and satisfyβ*0

*< β*1

*0;*→−

*ζ** _{c}* ζ1,c

*, ζ*2,c, with

*ζ*1,c

*>*0

*and*→−

*φ*^{}_{c}*being the eigenfunctions ofβ*0*andβ*1*, respectively. Moreover, the rest of the spectrum consists*
*of a discrete set of eigenvalues and the mappingc* →→−

*ζ*_{c}*is continuous with values inH*_{per}^{2} 0, L×
*H*_{per}^{1} 0, L.

**5. Concavity of** *dc*

* Lemma 5.1. Letc* ∈ −1,1

*andL > π*√

*2. Then the functiondcis concave, providedc*^{2} *<* 1/2
*and 1*−*c*^{2} *>*2π^{2}*/L*^{2}*.*

*Remark 5.2. Relation*1.8implies that*d*^{}c*<*0 is equivalent to the condition

*d*
*dcI*

*φ*_{wc}*, ψ*_{wc}*<*0. 5.1

*Proof ofLemma 5.1. Note that*

*d*
*dcI*

*φ*_{w}*, ψ*_{w}*d*
*dc*

_{L}

0

*φ*_{w}*ψ** _{w}*−

_{L}0

*φ*^{2}_{w}*dx*−*c* *d*
*dw*

_{L}

0

*φ*^{2}_{w}*dx*
*dw*

*dc*
−

_{L}

0

*φ*^{2}_{w}*dx*2c^{2} *d*
*dw*

_{L}

0

*φ*_{w}^{2}*dx*

*.* 5.2

Now,

*d*
*dw*

1 2

_{L}

0

*φ*_{w}^{2}*dx* 4
*L*

*d*

*dk*KkEk*dk*

*dw* *>*0. 5.3

Indeed, we observe from3.7,3.8, and3.11that
*φ**w*^{2}√

2η1

_{η}_{1}_{/}^{√}_{2L}

0

dn^{2}x;*kdx* 8Kk
*L*

_{K}

0

dn^{2}x;*kdx,* 5.4

where we used the fact that the Jacobi elliptic function dn has fundamental period 2Kand
is an even function. Now, by using that _{K}

0 cn^{2}x;*kdx* 1/k^{2}Ek −k^{}^{2}*Kk* and

dn^{2}x;*k *1−*k*^{2}*k*^{2}cn^{2}x;*k, it follows from*5.4that

1 2

_{L}

0

*φ*_{w}^{2}*dx* 4

*LKkEk.* 5.5

Now, Proposition 3.1and Theorem 3.2imply that the map *w* → Λw ≡ *η*2wis strictly
decreasing and from3.10, with*η*_{2} *η*_{2}w, we have that

*dk*
*dw* 1

2k

⎡

⎣2η_{2}^{2}−4wη_{2}*η*^{}_{2}
2w−*η*^{2}_{2} ^{2}

⎤

⎦*>*0. 5.6

Thus, since *k* ∈ 0,1 → *KkEk* is strictly increasing see Appendix, the claim 5.3
follows from5.5and5.6.

So, from5.2,5.3, and5.5, we get
*d*

*dcI*

*φ**w**, ψ**w* −8

*LKkEk * 16c^{2}
*L*

*d*

*dk*KkEk*dk*

*dw.* 5.7

Now, considering the functionΨdefined by2.12in12and using5.6, we obtain

*∂Ψ*

*∂w* 2√
2

2w−*η*^{2}_{2}dK/dw
*k*

*η*2*, w*

−2√ 2K

*k*

*η*2*, w*

2w−*η*^{2}_{2} ^{−1/2}
2w−*η*_{2}^{2}

2√ 2

2w−*η*^{2}_{2}dK/dk
*k*

*η*2*, w*
*η*^{2}_{2}*/k*

2w−*η*_{2}^{2} ^{2}
2w−*η*^{2}_{2}

−2√ 2K

*k*

*η*_{2}*, w*

2w−*η*^{2}_{2} ^{−1/2}
2w−*η*^{2}_{2} *,*

5.8

hence

*dk*

*dw* 1

2k

2w−*η*^{2}_{2} ^{2}

&

2η^{2}_{2}−4w
*k*

2w−*η*^{2}_{2} *K*−*η*^{2}_{2}dK/dk
*k*

2w−*η*_{2}^{2} *K*−2wdK/dk
'

*>*0. 5.9

From5.7,5.9, and using that 2w−*η*^{2}_{2} *η*^{2}_{2}*/k*^{}^{2}, we obtain*θL/8dI*φ*w**, ψ**w*/
*dc θ{−EK*2c^{2}E^{2}−*k*^{}^{2}*K*^{2}/kk^{}^{2}dk/dw}−θEK2c^{2}E^{2}−*k*^{}^{2}*K*^{2}Kη_{2}^{2}−2wk^{}^{2}
*K{−η*_{2}^{2}*Ek*^{2}*η*^{2}_{2}*K*−2wE2wk^{}^{2}*K}K{2c*^{2}E^{2}−*k*^{}^{2}*K*^{2}k^{}^{2}η^{2}_{2}−2w},or equivalently,

*θ* *L*
8K

*dI*
*φ**w**, ψ**w*

*dc* −η_{2}^{2}*E*

*k*^{2}*η*_{2}^{2}*K*−2wE2wk^{}^{2}*K*
2c^{2}

*k*^{} ^{2}

*E*^{2}−*k*^{}^{2}*K*^{2}

*η*^{2}_{2}−2w
*η*_{2}^{2}

−2wk^{}^{2}−*η*^{2}_{2}*k*^{2}
*EK*

2c^{2}

*k*^{} ^{2}*η*^{2}_{2}−4c^{2}*w*

*k*^{} ^{2}2wη^{2}_{2}
*E*^{2}
2c^{2}

*k*^{} ^{2}*η*^{2}_{2}*K*^{2}*,*

5.10

where*θη*^{2}_{2}k^{2}*η*^{2}_{2}*K*−2wE2wk^{}^{2}*K<*0.

Now, given thatk^{}^{2}*η*^{2}_{2}*/2w*−*η*^{2}_{2}, we rewrite the coeﬃcient of*E*^{2}in5.10as

2c^{2}

*k*^{} ^{2}*η*^{2}_{2}−4c^{2}*w*

*k*^{} ^{2}2wη^{2}_{2}2c^{2} *η*^{4}_{2}

2w−*η*^{2}_{2} −4c^{2}*w* *η*^{2}_{2}

2w−*η*^{2}_{2} 2wη_{2}^{2}
2c^{2}*η*^{4}_{2}−4c^{2}*wη*^{2}_{2}4w^{2}*η*^{2}_{2}−2wη^{4}_{2}

2w−*η*^{2}_{2}
*η*^{2}_{2}

2w−*η*_{2}^{2} 2w−2c^{2}

2w−*η*^{2}_{2} *η*_{2}^{2}

2w−2c^{2}
*.*

5.11

Also, the coeﬃcient of*EK*can be rewritten as*η*^{2}_{2}−2wk^{}^{2}−*η*_{2}^{2}*k*^{2} 2η_{2}^{4}*.*Thus,
*L*

8K
*dI*

*φ*_{w}*, ψ*_{w}

*dc* −2η^{4}_{2}*EK*2η^{2}_{2}

*w*−*c*^{2} *E*^{2}2c^{2}k^{}^{2}*η*^{2}_{2}*K*^{2}
*η*^{2}_{2}

*k*^{2}*η*^{2}_{2}*K*−2wE2wk^{}^{2}*K* *.* 5.12

We remark that we can write*w*as a function of complete elliptic integrals. In fact, by
integrating3.4from 0 to*L, we obtain*

*wwc *

_{L}

0 *φ*^{3}* _{w}*ξdξ

_{L}0 *φ**w*ξdξ*,* 5.13

which is well defined, since the solution*φ**w*is positive.

Now, using3.7, the expression 314.01 in11, and the fact that*Fπ/2;k Kk see*
Appendix, we obtain

_{L}

0

*φ** _{w}*ξdξ

_{L}0

*η*_{1}dn
*η*_{1}

√2*ξ;k*

*dξ*√
2

_{η}_{1}_{L/}^{√}_{2}

0

dn
*y;k* *dy*

√ 2

_{2K}

0

dn

*y;k* *dy*2√
2

_{K}

0

dn

*y;k* *dyπ*√
2.

5.14

Similarly using3.7, the expression 314.03 in11, and the special values sn00, and snK 1, cnK0see Appendix, it follows that

_{L}

0

*φ*^{3}* _{w}*ξdξ

_{L}0

*η*^{3}_{1}dn^{3}
*η*_{1}

√2*ξ;k*

*dξ*√
2η^{2}_{1}

_{2K}

0

dn^{3}
*y;k* *dξ*

2√
2η^{2}_{1}

_{K}

0

dn^{3}

*y;k* *dξ*16√
2*K*^{2}

*L*^{2}
1
2

1

*k*^{} ^{2}*π*

2 *k*^{2}snKcnK
4π√

2 1

*k*^{} ^{2}*K*^{2}
*L*^{2}*.*

5.15

Substituting5.14and5.15in5.13, we deduce that
*wc *1−*c*^{2} 4

1

*k*^{} ^{2}*K*^{2}

*L*^{2}*.* 5.16

Using5.16and*η*^{2}_{2}2wk^{}^{2}*/1* k^{}^{2}, the numerator of5.12will be positive if and only
if*c*^{2}k^{}^{2}*K*^{2}*>*c^{2}−*wE*^{2}*η*^{2}_{2}*EK*⇔1−*wk*^{}^{2}*K*^{2}*>*1−2wE^{2} 2wk^{}^{2}*/1* k^{}^{2}EK⇔
*w2E*^{2}−k^{}^{2}*K*^{2}−2k^{}^{2}*/1*k^{}^{2}EK*> E*^{2}−k^{}^{2}*K*^{2} ⇔*w >*E^{2}−k^{}^{2}*K*^{2}/2E^{2}−k^{}^{2}*K*^{2}−
2k^{}^{2}*/1k*^{}^{2}EK.

*Remark 5.3. 2E*^{2}−k^{}^{2}*K*^{2}−2k^{}^{2}*/1k*^{}^{2}EK > 0 since the functions*EK*and*EK*are
strictly increasingsee Appendix.

*claim 1.*

*k→*lim0

*E*^{2}−k^{}^{2}*K*^{2}
2E^{2}−k^{}^{2}*K*^{2}−

2k^{}^{2}*/1k*^{}^{2}

*EK* 2

5*.* 5.17

*Proof of claim. Indeed, denoting byf*k:*E*^{2}−k^{}^{2}*K*^{2}and by*gk*:*E*^{2}−2k^{}^{2}*/1k*^{}^{2}EK,
we use*L*^{}Hospital’s rule to find the limit5.17. Specifically, we show usingA.3that

*k*lim→0*f*^{j}k lim

*k→*0*g*^{j}k 0,

*j*0,1,2,3 *,*

*k*lim→0*f*^{4}k 3π^{2}

4 *,* lim

*k*→0*g*^{4}k 9π^{2}
8 *,*

5.18

which implies our claim.

Note that, by lim_{k}_{→1}*k*^{}^{2}*K*^{2}0, we have that

*k*lim→1

*E*^{2}−k^{}^{2}*K*^{2}
*E*^{2}−k^{}^{2}*K*^{2}*E*^{2}−

2k^{}^{2}*/1k*^{}^{2}

*EK* 1

2*.* 5.19

Moreover, 0*<*E^{2}−k^{}^{2}*K*^{2}/E^{2}−k^{}^{2}*K*^{2}E^{2}−2k^{}^{2}*/1k*^{}^{2}EK*<*1/2 for all*k*∈0,1,
since*E*^{2}−2k^{}^{2}*/1k*^{}^{2}K^{2}*> E*^{2}−*k*^{}^{2}*K*^{2} for all*k*∈0,1. In, addition we get1*k*^{}^{2}K >2E,
since the function*mk*: 1*k*^{}^{2}K−2Ehas the following properties:*m0 *0 and*m*^{}k*>*

0 for all*k*∈0,1. We conclude that the function
*fk*

*fk gk* *E*^{2}−k^{}^{2}*K*^{2}

*E*^{2}−k^{}^{2}*K*^{2}*E*^{2}−

2k^{}^{2}*/1k*^{}^{2}

*EK* 5.20

is strictly positive on 0,1. Now, continuity plus 5.17 and 5.19 implies that *c*_{0} :
max_{0≤k≤1}fk/fk *gk*satisfies 0*< c*_{0}≤1/2.

This concludes the lemma.

**6. Instability**

Consider the function*dc*defined by1.9. We now examine the relation between concavity
properties of*d*and the properties of the functional*H*near the critical point→−

*φ** _{c}*subject to the
constraint

*I*→−

*u I*→−

*φ** _{c}*.

**Theorem 6.1. Let**c /0 be fixed. Ifd^{}c *<* *0, then there exists a curvew* → →−

Φ*w* *which satisfies*
*I*→−Φ*w* *I*→−

*φ** _{c}*,→−Φ

*c*→−

*φ*_{c}*, and on whichH*→−*uhas a strict local maximum at*→−*u* →−
*φ*_{c}*.*
*Proof. We follow the ideas of* 3,8,9. Let→−

*ζ** _{c}* be the unique, negative eigenfunction of L

*c*. Define→−Φ

*w*:→−

*φ*_{w}*sw*→−

*ζ** _{c}*, for

*w*near

*c, wheresw*satisfies

*sc*0 and

*I*→−Φ

*w*

*I*→−

*φ*

*. The function*

_{c}*sw*can be defined by the implicit function theorem, since

*∂*

*∂sI*→−
*φ*_{w}*s*→−

*ζ*_{c}

_{{s0,wc}}

_{L}

0

*φ*_{c}*ζ*_{2,c}*ψ*_{c}*ζ*_{1,c} *dx,* 6.1

where→−

*ζ** _{c}* ζ1,c

*, ζ*

_{2,c}with

*ζ*

_{2,c}c/β0−1ζ1,c, and

*β*

_{0}is the unique negative eigenvalue of L

*c*and

*ψ*

*−cφ*

_{c}*c*,

*φ*

_{c}*>*0. Thus

*∂*

*∂sI*→−
*φ*_{w}*s*→−

*ζ*_{c}

_{{s0,wc}}−c

1 1

1−*β*0
*L*
0

*φ**c**ζ*1,c*dx /*0. 6.2

It is easy to see that

*d*^{2}
*dw*^{2}*I*→−

Φ*w*
{wc}

L*c*→−*y ,*→−*y*

*,* 6.3

where→−*y∂*→−

Φ*w**/∂w|*_{{wc}}*∂*→−

*φ*_{c}*/∂cs*^{}c→−

*ζ*_{c}*.*In fact, by some calculations, we have that
*d*^{}c −s^{}c

*I*^{}→−
*φ*_{c}

*,*→−
*ζ*_{c}

*,*
L*c*→−*y* −I^{}→−

*φ*_{c}

*s*^{}cL*c*→−
*ζ*_{c}*,*

6.4

so that

L*c*→−*y ,*→−*y*

−s^{}c
*I*^{}→−

*φ*_{c}*,*→−

*ζ*_{c}

*s*^{}c^{2}→−
*ζ*_{c}*,*L*c*→−

*ζ*_{c}

*<*0 6.5

in view of the fact that*d*^{}c*<*0.

To prove the instability, we need the following lemmas which are proved in8and as in the analogous case of7; therefore, we state them without proof.

**Lemma 6.2. There exist** >0 and a uniqueC^{1} map*α*:*U** _{}* → R, such that, for any→−

*u*u, v∈

*U*

_{}*andr*∈R,

→−*u*

·*α*→−*u* *, ∂** _{x}*→−

*φ*

_{c}0,

*α*→−*u·r* *α*→−*u* −*r,* *modulo the period,*

*α*^{}→−*u* *∂** _{x}*→−

*φ*

_{c}· −*α*→−*u*
→−*u, ∂*^{2}* _{x}*→−

*φ*_{c}

· −*α*→−*u* *,*

6.6

*whereU**is the “tube”*

*U*

"

−

→*u* ∈*X*: inf

*s∈R*

→−*u*−*τ**s*

→−
*φ*_{c}

*X**< *

#

*.* 6.7

*Definition 6.3. For*→−*u*∈*U*, define*B*→−*u*by the formula
*B*→−*u* →−*y*

· −*α*→−*u*

−*K∂*_{x}*α*^{}→−*u* *,* 6.8

where*K*_{0 1}

1 0 .