Orbital stability of periodic waves for the nonlinearSchrödinger equation

Thierry Gallay (1) , Mariana Haragus (2)

1 IF - Institut Fourier
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

The nonlinear Schrödinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.

Domaines: Equations aux dérivées partielles [math.AP]

Thierry Gallay, Mariana Haragus. Orbital stability of periodic waves for the nonlinearSchrödinger equation. Journal of Dynamics and Differential Equations, 2007, 19, pp.825-865. ⟨hal-00351501⟩