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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 solitons3,4, 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 solutionsφxctof the modified Boussinesq equation

uttuxx

u3uxx

xx0. 1.1

The above equation1.1, has the following equivalent form as a Hamiltonian system utvx,

vt

uuxxu3

x

1.2

(2)

forx∈R,t >0. Here subscriptstandxdenote partial differentiation with respect totandx.

The above equation conserves energy, namely, the integral

Hu, v 1 2

L

0

u2v2u2xu4 2

dx 1.3

does not depend on the timet. 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, letτsbe the translation bys,τsφx φxsforx∈Rand let→−

φc φcx−ct, ψcx−ctbe anL-periodic travelling wave solution to system1.2, whereφc:R → R,ψc:R → R,L >0 is the period ofφcandψc, andcis the wave’s speed of propagation. If we define the→−

φc-orbit to be the setΩφ

c {→−

φc·s, s∈R},

φcis called orbitally stable if profiles near its orbit remain near the orbit for as long as it exists.

So, we have the following definition. LetXbe a Hilbert space.

Definition 1.1 Orbital Stability. Let→−

φc φcx −ct, ψcx−ctX be an L-periodic travelling wave solution to system1.2. We say that the orbit Ωφ

c is stable in theX-sense by the flow of system1.2if for each >0 there existsδ δ>0 such that if→−u0Xand infs∈R−→u0τs→−

φcX < δ,then the solution→−utof1.2with→−u0 →−u0satisfies, for alltfor which→−u u, vexists,

infs∈R

→−utτs

→− φc

X < . 1.5

Otherwise, we say thatΩφ

c isX-unstable.

Here,X : Hper1 0, L×L2per0, L.The choice of norm in1.5is dictated by the form of the Hessian or “linearized Hamiltonian”H→−

φc cI→−

φcand varies from problem to problem.

Inserting theL-periodic travelling wave solution→−

φc φcx−ct, ψcx−ctin1.2 leads to the system

−cφcξ ψcξ,

−cψcξ

φcφcφc3

ξ, 1.6

(3)

where ’ connotesd/dξandξ xct. Integrating the latter system, we obtain the nonlinear system

−cφξ ψξ K1,

−cψξ φξφξ−φ3ξ K2,

1.7

whereK1, K2are integration constants, which will be considered equal to zero here. Then, we obtain

HcI −→ φc

0. 1.8

Next observe that relation1.8characterizes→−

φc φc, ψcas a critical point ofHsubject to the constraintIu, v Iφc, ψc.In order to prove instability for→−

φc, we will examine the relation between the concavity properties of the function

dc H→− φc·

cI→− φc·

, 1.9

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

φc under the constraintI constant.

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

φ φcx−ct, ψcx−

ctof the generalized Boussinesq equation utvx,

vt uuxxupx 1.10

are stable in theH1L2Rnorm for speedscsuch thatp−1/4< c2<1 ifdin1.9is a convex function ofc. The aim of this paper is to prove that the solutions given byTheorem 3.2 below are unstable ifdcis 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 thex-variable for the system1.10for everyp. For this reason, we will treat here only the casep 3. Stability of dnoidal waves for this case is also treated by the author in a forthcoming paper6. Regarding the classical casep2, in5the author proves nonlinear stability properties of a class ofL-periodic travelling wave solutions, called cnoidal waves, in the energy spaceHper1 0, L×L2per0, L, by periodic disturbances with periodL.

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

φc φc, ψc of dnoidal wave solutions to system1.2, with a fixed periodLTheorem 3.2below. Then, a proof of orbital instability of these solutions is established inX 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→−

φc, given by Theorem 3.2below, is the following.

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Theorem 1.2Instability Theorem. Letc ∈ −1,1and L > π

2. Then the orbitΩφ

c is X-

unstable with respect to the flow of the modified Boussinesq equation, providedc2 <1/2 and 1−c2 >

2/L2.

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{→−

φc}c on the parameterc, of positive dnoidal wave solutions to system 1.2, with a fixed period LTheorem 3.2belowand inSection 4we present a complete study of the spectrum of the operatorLc. The existence of the smooth curvec →→−

φc will allow us to differentiate the functiondc. Then, in section 5, we prove thatdcis indeed concave, for a certain range of speeds and periods of→−

φc, 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 periodLinX.

The following notation will be used:

f, g

0

f, g

L2per0,L L

0

fgdx, f, g

1

f, g

Hper0,L1 L

0

fgdx L

0

fgdx, f

0f

L2per0,L L

0

f2dx 1/2

, f

1f

Hper0,L1 L

0

f2dx L

0

f2dx 1/2

, f, g ,u, v

f, g ,u, v

L2per0,L×L2per0,L L

0

fudx L

0

gvdx, f, g f, g L2

per0,L×L2per0,L L

0

f2dx L

0

g2dx 1/2

, f, g Xf, g H1

per0,L×L2per0,L L

0

f2dx L

0

f2dx L

0

g2dx 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→−u0 u0, v0XHper1 0, L×L2per0, L. Then there existT > 0 and a uniquely weak solution→−u u, vof 1.2with→−u0 →−u0.

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Proof. In order to obtain the existence of weak solutions for the system1.2, we consider the approximate problem

utA→−uF→−u ,

u0 →−un0, 2.1

with→−un0DA Hper3 0, L×Hper1 0, Land→−un0 → −→u0inX, where

A

0 −∂x

−∂x3x 0

2.2

and −A is the infinitesimal generator of a C0 group of unitary operators in X and F Ft, u, v

−∂x0u3

. SinceFC, the map u, v → 0, ∂xu3 is locally Lipschitz on X. But then for all→−un0DA, there exists aTn >0 such that the initial value problem2.1 has a unique solution→−unC0, Tn;DAC10, Tn;X. Moreover, ifTn<∞, then

t→limTn

→−unt

X ∞, 2.3

by the semigroup theory10. By2.1, we estimate on0, Tn 1

2 d dt

→−unt2

X →−unt, ∂t→−unt

X →−un,−A→−unF→−un

X

L

0

xun3vntdx

≤∂xun3t

0vnt0 3

L

0

unt2untx2dx 1/2

vnt0

≤3unt2 L

0

unxt2dx 1/2

vnt0

≤3SLunt21unx0vnt0

≤3SL→−unt2

X

→−un2

X,

2.4

where we used in the first equality above that−A→−unX.

Consider nowfs 3SLs, which is a continuous, positive and increasing function onR. Then by Gronwall’s inequality, it follows that

→−unt2

X≤→−un02

Xexp t

0

f

→−unτ2

X

, on0, Tn. 2.5

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We compare−→unt2Xwith the maximal solutionyt

yt≡ 1

supn→−un0

X

2

1−3St

supn→−un0

X

2,

t∈0, T0

⎢⎣0, 1 3S

supn→−un0

X

2

⎟⎠

2.6

of the scalar Cauchy problem

dy dt f

y y, y0 y0sup

n

→−un02

X.

2.7

It follows that

→−unt2

Xyt, on0, Tn∩0, T0. 2.8

LetT < T0. Then→−unis defined on0, Tfor alln. Moreover, →−unt2

XC0y0 K2 2.9

on0, T, whereKis a constant independent ofn, since by2.5,2.8, and the fact thatyt is bounded on0, T, we have the following inequality on0, T:

→−unt2

X ≤→−un02

Xexp t

0

f

→−unτ2

X

≤→−un02

Xexp T

0

f

C0T→−un02

XC0Ty0.

2.10

Finally, from2.9 and standard weak limit arguments, we have the existence of a unique solution→−utC0, T;X.

Proposition 2.2. The unique solution→−utof 1.2with initial data→−u0 →−u0, 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.

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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

φxct, ψxct . 3.1

Substituting3.1in1.2leads to the system

−cφξ ψξ,

−cψξ

φφφ3

ξ, 3.2

where ’ denotesd/dξandξxct. Integrating3.2, we obtain the nonlinear system

−cφξ ψξ K1,

−cψξ φξφξ−φ3ξ K2, 3.3

whereK1, K2are integration constants, which will be considered equal to zero here. Then,φ must satisfy

φwφφ30, 3.4

wherewwc 1−c2will be considered positive.

Next, we show how to construct a smooth curve of solutions for 3.4 with a fixed fundamental periodL, and depending on the parameterc. In order to do this, we first observe from3.4thatφsatisfies the first-order equation

φ 2 1 2

−φ42wφ24Bφ

1 2

η21φ2

φ2η22

, 3.5

whereBφ is an integration constant and−η1,η1,−η2,η2are the real zeros of the polynomial pφt −t42wt24Bφ, which satisfy the relations

2wη21η22,

4Bφ −η12η22. 3.6

Moreover, we assume without lost of generality thatη1 > η2 > 0 and we obtain from3.5 thatη2φη1. By definingϕφ/η1andk2 η21η2221,3.5becomesϕ2 η21/21ϕ2ϕ2−1k2. We also impose the crest of the wave to be atξ 0, that is,φ0 1. Now, we define a further variableψ via the relationϕ2 1−k2sin2ψ and so we get thatψ2 η12/21k2sin2ψ.Then we obtain for1/

2 thatψξ

0 dt/

1−k2sin2t lξ.Therefore,

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from the definition of the Jacobian elliptic functiony snu;k see in the appendix or in Byrd and Friedman11, we can write the last equality as sinψsnlξ;kand henceϕξ 1−k2sn2lξ;k dnlξ;k.Returning to the initial variable, we obtain the called dnoidal wave solution associated to3.4,

φξφ

ξ;η1, η2 η1dn η1

√2ξ;k

3.7

with

k2 η21η22

η21 , η12η222w, 0< η2< η1. 3.8

Next, dn has fundamental period 2K, dnu2K;k dnu;k, whereK Kkrepresents the complete elliptic integral of the first kindsee appendix; it follows that the dnoidal wave φin3.7has fundamental period,Tφ, given by

Tφ≡ 2√ 2

η1 Kk. 3.9

Now, we show thatTφ > √ 2π/√

w. First, we expressTφ as a function ofη2 andw. In fact, for everyη2 ∈0,√

w, there is a uniqueη1 ∈√ w,

2wsatisfying the first relation in3.6, namely,η1

2w−η22. So, from3.9we obtain

Tφ

η2, w 2√ 2

2w−η22 K

k η2

, withk2

η2, w 2w−2η22

2w−η22 . 3.10

Then, by fixingw >0, we have thatTφ → ∞asη2 → 0 andTφη2 → π√ 2/√

w as η2 → √

w. So, since the mapping η2Tφwη2 is strictly decreasing see proof of Proposition 3.1, it follows thatTφ>

2π/√ w.

Now, we obtain a dnoidal wave solution with periodL. Forw0 >2/L2, there is a uniqueη2,0 ∈ 0,√

w0such thatTφη2,0, w0 L. So, forη1,0such thatη1,02 η22,0 2w0, the dnoidal waveφ· φ·, η1,0, η2,0has a fundamental periodLand satisfies3.4withww0. By the above analysis the dnoidal waveφ·, η1, η2in3.7is completely determined bywandη2and will be denoted byφw·;η2orφw.

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. LetL > 0 be arbitrary but fixed. Considerw0 >2/L2 and the uniqueη2,0 η2w0∈0,√

w0such thatTφw

0 L. Then,

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1there exist an interval Iw0 aroundw0, an interval 2,0around η2,0, and a unique smooth functionΛ:Iw02,0such thatΛw0 η2,0and

2√ 2

2w−η22

Kk L, 3.11

wherew∈ Iw0,η2 Λw, andk2k2w∈0,1is defined by3.10;

2the positive dnoidal wave solution in 3.7, φw·;η1, η2, determined by η1η1w 2w−η22, η2η2w, has fundamental period L and satisfies 3.4. Moreover, the mappingw∈ Iw0φwHper1 0, Lis a smooth function;

3Iw0can be chosen as2/L2,∞;

4the mappingΛ:2π2/L2,∞ → 2,0is 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 system1.2inHpern 0, L×Hperm 0, L,n, m0 which satisfy the system3.3with integration constantsK1K20; this curve is given, forwc 1−c2, by

c

⎝−

! 1−2π2

L2 ,

! 1− 2π2

L2

⎠−→

φwc, ψwc . 3.12

Moreover,φcξ:φwcξ

2w−η22dn

2w−η22/

2ξ;k, ψwc −cφwc,where the smooth functionη2η2wcis given byProposition 3.1andkkwcby3.10.

Remark 3.3. →−

φc/∂cis inHper0, L×Hper0, Las soon as inHper1 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 LcH

φwc, ψwc cI

φwc, ψwc 4.1

determined by the periodic solutionsφwc, ψwc found in Theorem 3.2. We compute the Hessian operatorLcby calculating the associated quadratic form, which is denoted byQc. By definition,Qcg, his the coefficient of2in

H

φwcg, ψwch cI

φwcg, ψwch , 4.2

(10)

and so is given by

Qc

g, h L

0

"

1 2

g2gx2h2

−3

2φ2wcg2cgh

# dx

L

0

"

1 2

1−c2

g2g2 −3φ2wcg2 1

2

hcg 2dx

#

:Q1c g 1

2hcg2

0.

4.3

Note thatQcis the sum of the quadratic formQ1cassociated to the operator−d2/dx21−c2− 3φ2wand the nonnegative term1/2hcg20. From3.2for the dnoidal waveφwc, ψwc, it follows that g φwc and h ψwc satisfy Lcg, h 0. To see that this is the only eigenfunction corresponding to the eigenvalue zero and the other expected properties of the operatorLc, we will first consider the following result about the periodic eigenvalue problem:

Ldnξ:

d2

dx2 w−3φ2w

ξλ, ξ0 ξL, ξ0 ξL,

4.4

whereφwis given byProposition 3.1.

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

Theorem 4.1. LetLdn be the linear operator defined onHper2 0, Lby 4.4. Then the first three eigenvaluesβ12, andβ3ofLdnare simple, and satisfyβ1<0β2< β3, andφis the eigenfunction ofβ2. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues which are double.

To prove that the kernel of Lc is spanned by d/dxφwc, ψwc, consider the quadratic formQcg, has the pairing ofg, hagainstg,hin theHper1 0, L×L2per0, L− Hper−10, L×L2per0, Lduality, whereg,h tis the unbounded operator

Lc:

1−xx−3φ2wc c

c 1

4.5

applied tog, ht. ThenLcg, ht0 implies

−g 1−c2

g−3φwc2 g 0, h−cg.

4.6

Now, from the properties of the operatorLdn −∂2xw−3φw2 established inTheorem 4.1, it follows thatgλφwc andh−cg −cλφwcλψwc , whereλ /0∈R.

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To show that there is a single, simple, negative eigenvalue, considerQ1c defined in 4.3above. ByTheorem 4.1, the operatorLdn has exactly one negative eigenvalue which is simple, sayλ0, with associated eigenfunctionζ > 0. Thus,Q1cachieves a negative value and so doesQc. In fact, considering→−

ζ ζ,−cζ, we haveLc→−

ζ Q1cζ 1/2cζ−2 Q1cζ 1/2λ0<0.Denoting byβ0the lowest eigenvalue ofLc, we will show that the next eigenvalueβ1is 0, which is known to be simple, and consequentlyβ2is in fact strictly positive.

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

β1 max

φ11∈X min g,h∈X\{0}

g,φ11h,ψ100

Qc g2g, h

1h20. 4.7

Choosingφ1ζ,ψ10, we obtain the lower estimate

β1 ≥ min

g,h∈X\{0}

g,ζ10

Qc

g, h g2

1h20. 4.8

The right-hand side of4.8is nonnegative on the subspace{g, h ∈ X\ {0};g, ζ1 0}, sinceQ1cg ≥ 0 byTheorem 4.1. Thus,β1 0 and, from earlier considerations,β1 is simple andβ2>0.

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

Theorem 4.2. LetLcbe the linear operator defined onHper2 0, L×Hper1 0, Lby4.1. Then the first two eigenvaluesβ0andβ1ofLcare simple and satisfyβ0< β10;→−

ζc ζ1,c, ζ2,c, withζ1,c>0 and→−

φcbeing the eigenfunctions ofβ0andβ1, respectively. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues and the mappingc →→−

ζcis continuous with values inHper2 0, L× Hper1 0, L.

5. Concavity of dc

Lemma 5.1. Letc ∈ −1,1andL > π

2. Then the functiondcis concave, providedc2 < 1/2 and 1c2 >2/L2.

Remark 5.2. Relation1.8implies thatdc<0 is equivalent to the condition

d dcI

φwc, ψwc <0. 5.1

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Proof ofLemma 5.1. Note that

d dcI

φw, ψw d dc

L

0

φwψwL

0

φ2wdxc d dw

L

0

φ2wdx dw

dc

L

0

φ2wdx2c2 d dw

L

0

φw2dx

. 5.2

Now,

d dw

1 2

L

0

φw2dx 4 L

d

dkKkEkdk

dw >0. 5.3

Indeed, we observe from3.7,3.8, and3.11that φw2

1

η1/2L

0

dn2x;kdx 8Kk L

K

0

dn2x;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 cn2x;kdx 1/k2Ek −k2Kk and

dn2x;k 1−k2k2cn2x;k, it follows from5.4that

1 2

L

0

φw2dx 4

LKkEk. 5.5

Now, Proposition 3.1and Theorem 3.2imply that the map w → Λw ≡ η2wis strictly decreasing and from3.10, withη2 η2w, we have that

dk dw 1

2k

⎣2η22−4wη2η2 2w−η22 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

φw, ψw −8

LKkEk 16c2 L

d

dkKkEkdk

dw. 5.7

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

∂Ψ

∂w 2√ 2

2w−η22dK/dw k

η2, w

−2√ 2K

k

η2, w

2w−η22 −1/2 2w−η22

2√ 2

2w−η22dK/dk k

η2, w η22/k

2w−η22 2 2w−η22

−2√ 2K

k

η2, w

2w−η22 −1/2 2w−η22 ,

5.8

hence

dk

dw 1

2k

2w−η22 2

&

22−4w k

2w−η22 Kη22dK/dk k

2w−η22 K−2wdK/dk '

>0. 5.9

From5.7,5.9, and using that 2w−η22 η22/k2, we obtainθL/8dIφw, ψw/ dc θ{−EK2c2E2k2K2/kk2dk/dw}−θEK2c2E2k2K222−2wk2 K{−η22Ek2η22K−2wE2wk2K}K{2c2E2k2K2k2η22−2w},or equivalently,

θ L 8K

dI φw, ψw

dc −η22E

k2η22K−2wE2wk2K 2c2

k 2

E2k2K2

η22−2w η22

−2wk2η22k2 EK

2c2

k 2η22−4c2w

k 22wη22 E2 2c2

k 2η22K2,

5.10

whereθη22k2η22K−2wE2wk2K<0.

Now, given thatk2η22/2wη22, we rewrite the coefficient ofE2in5.10as

2c2

k 2η22−4c2w

k 22wη222c2 η42

2w−η22 −4c2w η22

2w−η22 2wη22 2c2η42−4c2224w2η22−2wη42

2w−η22 η22

2w−η22 2w−2c2

2w−η22 η22

2w−2c2 .

5.11

Also, the coefficient ofEKcan be rewritten asη22−2wk2η22k224.Thus, L

8K dI

φw, ψw

dc −2η42EK22

wc2 E22c2k2η22K2 η22

k2η22K−2wE2wk2K . 5.12

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We remark that we can writewas a function of complete elliptic integrals. In fact, by integrating3.4from 0 toL, we obtain

wwc

L

0 φ3wξdξ L

0 φwξdξ, 5.13

which is well defined, since the solutionφwis positive.

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

L

0

φwξdξ L

0

η1dn η1

√2ξ;k

√ 2

η1L/2

0

dn y;k dy

√ 2

2K

0

dn

y;k dy2√ 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

φ3wξdξ L

0

η31dn3 η1

√2ξ;k

√ 2η21

2K

0

dn3 y;k

2√ 2η21

K

0

dn3

y;k 16√ 2K2

L2 1 2

1

k 2π

2 k2snKcnK 4π√

2 1

k 2K2 L2.

5.15

Substituting5.14and5.15in5.13, we deduce that wc 1−c2 4

1

k 2K2

L2. 5.16

Using5.16andη222wk2/1 k2, the numerator of5.12will be positive if and only ifc2k2K2>c2wE2η22EK⇔1−wk2K2>1−2wE2 2wk2/1 k2EK⇔ w2E2−k2K2−2k2/1k2EK> E2−k2K2w >E2−k2K2/2E2−k2K2− 2k2/1k2EK.

Remark 5.3. 2E2−k2K2−2k2/1k2EK > 0 since the functionsEKandEKare strictly increasingsee Appendix.

claim 1.

k→lim0

E2−k2K2 2E2−k2K2

2k2/1k2

EK 2

5. 5.17

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Proof of claim. Indeed, denoting byfk:E2−k2K2and bygk:E2−2k2/1k2EK, we useLHospital’s rule to find the limit5.17. Specifically, we show usingA.3that

klim0fjk lim

k→0gjk 0,

j0,1,2,3 ,

klim→0f4k 3π2

4 , lim

k0g4k 9π2 8 ,

5.18

which implies our claim.

Note that, by limk→1k2K20, we have that

klim→1

E2−k2K2 E2−k2K2E2

2k2/1k2

EK 1

2. 5.19

Moreover, 0<E2−k2K2/E2−k2K2E2−2k2/1k2EK<1/2 for allk∈0,1, sinceE2−2k2/1k2K2> E2k2K2 for allk∈0,1. In, addition we get1k2K >2E, since the functionmk: 1k2K−2Ehas the following properties:m0 0 andmk>

0 for allk∈0,1. We conclude that the function fk

fk gk E2−k2K2

E2−k2K2E2

2k2/1k2

EK 5.20

is strictly positive on 0,1. Now, continuity plus 5.17 and 5.19 implies that c0 : max0≤k≤1fk/fk gksatisfies 0< c0≤1/2.

This concludes the lemma.

6. Instability

Consider the functiondcdefined by1.9. We now examine the relation between concavity properties ofdand the properties of the functionalHnear the critical point→−

φcsubject to the constraintI→−u I→−

φc.

Theorem 6.1. Letc /0 be fixed. Ifdc < 0, then there exists a curvew → →−

Φw which satisfies I→−Φw I→−

φc,→−Φc→−

φc, and on whichH→−uhas a strict local maximum at→−u →− φc. Proof. We follow the ideas of 3,8,9. Let→−

ζc be the unique, negative eigenfunction of Lc. Define→−Φw:→−

φwsw→−

ζc, forwnearc, whereswsatisfiessc 0 andI→−Φw I→− φc. The functionswcan be defined by the implicit function theorem, since

∂sI→− φws→−

ζc

{s0,wc}

L

0

φcζ2,cψcζ1,c dx, 6.1

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where→−

ζc ζ1,c, ζ2,cwithζ2,c c/β0−1ζ1,c, andβ0is the unique negative eigenvalue of Lcandψc−cφc,φc>0. Thus

∂sI→− φws→−

ζc

{s0,wc}−c

1 1

1−β0 L 0

φcζ1,cdx /0. 6.2

It is easy to see that

d2 dw2I→−

Φw {wc}

Lc→−y ,→−y

, 6.3

where→−y∂→−

Φw/∂w|{wc}→−

φc/∂csc→−

ζc.In fact, by some calculations, we have that dc −sc

I→− φc

,→− ζc

, Lc→−y −I→−

φc

scLc→− ζc,

6.4

so that

Lc→−y ,→−y

−sc I→−

φc ,→−

ζc

sc2→− ζc,Lc→−

ζc

<0 6.5

in view of the fact thatdc<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 uniqueC1 mapα:U → R, such that, for any→−u u, v∈U andr∈R,

→−u

·α→−u , ∂x→− φc

0,

α→−u·r α→−ur, modulo the period,

α→−u x→− φc

· −α→−u →−u, ∂2x→−

φc

· −α→−u ,

6.6

whereUis the “tube”

U

"

uX: inf

s∈R

→−uτs

→− φc

X<

#

. 6.7

Definition 6.3. For→−uU, defineB→−uby the formula B→−u →−y

· −α→−u

K∂xα→−u , 6.8

whereK0 1

1 0 .

参照

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