Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

Erwan Faou; Ludwig Gauckler; Christian Lubich

hal-00622240, version 2

Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

Erwan Faou () ^1, 2, Ludwig Gauckler () ³, Christian Lubich () ⁴

(2011)

Résumé : It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.

1 : Institut de Recherche Mathématique de Rennes (IRMAR)
CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
2 : IPSO (INRIA - IRMAR)
CNRS : UMR6074 – INRIA – Université de Rennes 1
3 : Institut für Mathematik [Berlin]
Technische Universität Berlin
4 : Mathematisches Institut [Tubigen]
Universität Tübingen

hal-00622240, version 2
http://hal.archives-ouvertes.fr/hal-00622240
oai:hal.archives-ouvertes.fr:hal-00622240
Contributeur : Marie-Annick Guillemer <>
Soumis le : Jeudi 11 Octobre 2012, 14:38:53
Dernière modification le : Jeudi 11 Octobre 2012, 21:13:41

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arXiv : 1109.2407

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