Stability of small periodic waves for the nonlinear Schrodinger equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:234

Article

Gallay, Thierry ; Haragus, Mariana

关键词: nonlinear Schrodinger equation;    periodic waves;    orbital stability;    spectral stability;   

DOI  :  10.1016/j.jde.2006.12.007

来源: Elsevier

【 摘 要 】

The nonlinear Schrodinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling 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 is an element of R. In this paper we investigate the stability of the small amplitude traveling 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 traveling 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. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

Free