Stability of small periodic waves for the nonlinear Schrodinger equation

Thierry Gallay

Description

Niveau: Supérieur, Doctorat, Bac+8
Stability of small periodic waves for the nonlinear Schrodinger equation Thierry Gallay Institut Fourier Universite de Grenoble I B.P. 74 38402 Saint-Martin-d'Heres, France Mariana Ha˘ra˘gus¸ Departement de Mathematiques Universite de Franche-Comte 16 route de Gray 25030 Besanc¸on, France Abstract The nonlinear Schrodinger equation possesses three distinct six-parameter families of complex- valued quasi-periodic travelling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x?ct for some c ? R. In this paper we investigate the stability of the small amplitude travelling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude travelling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability. Running head: Periodic waves in the NLS equation Corresponding author: Mariana Ha˘ra˘gus¸, Keywords: nonlinear Schrodinger equation, periodic waves, orbital stability, spectral stability

• wave


• spectral stability


• nonlinear stability


• periodic solution


• waves reduces


• periodic waves


• has also


• nls equation


• cubic nonlinear

Sujets

• Periodic function

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 6
• Langue: English

Extrait

Stability

of

small periodic waves for the Schrodingerequation

Thierry Gallay Institut Fourier UniversitedeGrenobleI B.P. 74 38402Saint-Martin-d’Heres,France

Mariana Haragus DepartementdeMathematiques UniversitedeFranche-Comte 16 route de Gray 25030Besancon,France

Abstract

nonlinear

The nonlinear Schrodinger equation possesses three distinct six-parameter families of complex- valued quasi-periodic travelling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function ofx ctfor some c∈Rthis paper we investigate the stability of the small amplitude travelling waves, both . In in the defocusing and the focusing case. Our rst result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude travelling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.

Running head:Periodic waves in the NLS equation

Corresponding author:Mariana Haragus,octm.erfarhf-vinu.htam@suga

Keywords:tsbatila,ypslitialstectrityabilnonlinearSchrognidqereitaup,noioercwdiesavrb,o

1 Introduction

Weconsidertheone-dimensionalcubicnonlinearSchrodingerequation(NLS) iU t (x, t) +U xx (x, t) |U(x, t)| 2 U(x, t) = 0, wherex∈R,t∈R,U(x, t)∈C, and the signs + and in the nonlinear term correspond to the focusing and the defocusing case, respectively. In both cases the NLS equation possesses quasi-periodicsolutions of the general form U(x, t) = e i(px ωt) V(x ct), x∈R, t∈R,(1.1)

wherep, ω, caererpaalmerarstedtanleoprvewaheVis acomplex-valued periodicfunction of its argument. The aim of the present paper is to investigate the stability properties of these particular solutions,atleastwhenthewaveproleV It turns out is small.that the discussion is very similar in both cases, so for simplicity we restrict our presentation to the defocusing equation iU t (x, t) +U xx (x, t) |U(x, t)| 2 U(x, t) = 0,(1.2) and only discuss the dierences which occur in the focusing case at the end of the paper.

A crucial role in the stability analysis is played by the various symmetries of the NLS equation. The most important ones for our purposes are the four continuous symmetries: phase invariance:U(x, t)7→U(x, t) e iϕ ,ϕ∈R;

translation invariance:U(x, t)7→U(x+ , t),∈R; Galilean invariance:U(x, t)7→e  i ( 2 x+ 2 4 t ) U(x+vt, t),v∈R; dilation invariance:U(x, t)7→U(x,  2 t), >0;

and the two discrete symmetries:

reectionsymmetry:U(x, t)7→U( x, t), conjugation symmetry:U(x, t)7→U(x, t).

As is well-known, the Cauchy problem for equation (1.2) is globally well-posed on the whole real line in the Sobolev spaceH 1 (R,C one can solve the Alternatively, 13, 14, 21]. [9,), see e.g. NLS equation on a bounded interval [0, L] with periodic boundary conditions, in which case an appropriate function space isH 1 rpe ([0, L],C). In both situations, we have the following conserved quantities: E 1 (U 1 2 )= Z I |U(x, t)| 2 dx , i E 2 (U  2 ) = Z I U(x, t)U x (x, t) dx , E 3 (U) = Z I  2 1 |U x (x, t)| 2 1 4 +|U(x, t)| 4  dx ,

2

whereIdenotes either the whole real line or the bounded interval [0, L quantities]. TheE 1 and E 2 are conserved due to the phase invariance and the translation invariance, respectively, whereas the conservation ofE 3 originates in the fact that equation (1.2) is autonomous. The symmetries listed above are also useful to understand the structure of the set of all quasi- periodic solutions of (1.2). Assume thatU(x, t) is a solution of (1.2) of the form (1.1), where V:R→C Sinceis a bounded function.|U(x, t)|=|V(x ct)|, the translation speedc∈Ris uniquely determined byU, except if the modulus|V| any case, using the Galileanis constant. In invariance, we can transformU(x, tinto another solution of the form (1.1) with) c this Once= 0. is done, the temporal frequencyωis in turn uniquely determined byU(x, t), except in the trivial case whereV Inis identically zero. view of the dilation invariance, only the sign ofωis important, so we can assume without loss of generality thatω∈ {1; 0; 1}. SettingU(x, t) = e  iωt W(x), we see thatW(x) = e ipx V(x) is a bounded solution of the ordinary dieren tial equation W xx (x) +ωW(x) |W(x)| 2 W(x) = 0, x∈R.(1.3) Ifω= 0 orω= 1, is it straightforward to verify thatW0 is the only bounded solution of (1.3), thus we assume from now on thatω (1.3) is actually the stationary Ginzburg-= 1. Equation Landau equation and the set of its bounded solutions is well-known [6, 10, 11, 12]. There are two kinds of solutions of (1.3) which lead to quasi-periodic solutions of the NLS equation of the form (1.1):

A family ofperiodic solutionswith constant modulusW(x) = (1 p 2 ) 1/2 e i(px+ϕ) , where p∈[ 1,1] andϕ∈[0,2 corresponding solutions of (1.2) are called]. Theplane waves. The general form of these waves is U(x, t) = e i(px ωt) V , wherep∈R,ω∈R, andV∈Csatisfy the dispersion relationω=p 2 +|V| 2 . A family ofquasi-periodic solutionsof the formW(x) =r(x) e iϕ(x) , where the modulusr(x) and the derivative of the phaseϕ(x Any) are periodic with the same period. such solution can be written in the equivalent formW(x) = e ipx Q(2kx), wherep∈R,k >0, andQ:R→C is 2-periodic. In particular, U(x, t) = e  it W(x) = e i(px t) Q(2kx) (1.4)

is a quasi-periodic solution of (1.2) of the form (1.1) (withc= 0 andω= 1). We shall refer to such a solution as aperiodic wave, because its prole|U(x, t)|is a (non-trivial) periodic function of the space variablex quantities related to the periodic wave (1.4) are. Important the period of the modulusT= /k, and the Floquet multiplier e ipT  small amplitude. For solutions (|Q| 1) the minimal periodTis close to, hencek1, and the Floquet multiplier is close to 1, so that we can choosep1.

While the plane waves form a three-parameter family, we will see in Section 2 that the periodic waves form a six-parameter family of solutions of (1.2). However, the number of independent parameters can be substantially reduced if we use the four continuous symmetries listed above. Indeed it is easy to verify that any plane wave is equivalent either toU 1 (x, t) = 0 or toU 2 (x, t) = e  it  a similar way, the set of all periodic waves reduces to a two-parameter family.. In

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