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Orbital stability of periodic waves for the nonlinear Schrödinger equation

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The nonlinear Schrödinger equation has several families of quasi-periodic traveling 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 (Gallay and Haragus, J. Diff. Equations, 2007) 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.

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Authors and Affiliations

  1. Institut Fourier, Université de Grenoble I, B.P. 74, 38402, Saint-Martin-d’Hères, France

    Thierry Gallay

  2. Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030, Besançon, France

    Mariana Hǎrǎgus

Authors
  1. Thierry Gallay
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  2. Mariana Hǎrǎgus
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Correspondence to Thierry Gallay.

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Gallay, T., Hǎrǎgus, M. Orbital stability of periodic waves for the nonlinear Schrödinger equation. J Dyn Diff Equat 19, 825–865 (2007). https://doi.org/10.1007/s10884-007-9071-4

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  • DOI: https://doi.org/10.1007/s10884-007-9071-4

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