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

Erwan Faou; Ludwig Gauckler; Christian Lubich

doi:10.1080/03605302.2013.785562

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

* Corresponding author

1 IPSO - Invariant Preserving SOlvers

IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique

2 IRMAR - Institut de Recherche Mathématique de Rennes

3 Institut für Mathematik [Berlin]

4 Mathematisches Institut [Tubigen]

Abstract : 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.

Keywords : Nonlinear Schrödinger equation Normal form Modulated Fourier expansion

Document type :

Journal articles

Communications in Partial Differential Equations, Taylor & Francis, 2013, 38 (7), pp.1123-1140. <10.1080/03605302.2013.785562>

Domain :

Mathematics [math] / Analysis of PDEs [math.AP]

https://hal.archives-ouvertes.fr/hal-00622240
Contributor : Marie-Annick Guillemer <>
Submitted on : Thursday, October 11, 2012 - 2:38:53 PM
Last modification on : Thursday, July 7, 2016 - 11:31:21 AM

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

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