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PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION

Part of: Numerical problems in dynamical systems

Published online by Cambridge University Press:  10 April 2014

ERWAN FAOU ,
LUDWIG GAUCKLER  and
CHRISTIAN LUBICH
ERWAN FAOU
Affiliation:
INRIA and ENS Cachan Bretagne, Avenue Robert Schumann, F-35170 Bruz, FranceErwan.Faou@inria.fr Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, France
LUDWIG GAUCKLER
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germanygauckler@math.tu-berlin.de
CHRISTIAN LUBICH
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germanylubich@na.uni-tuebingen.de

Abstract

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Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.

MSC classification

Secondary: 65P10: Hamiltonian systems including symplectic integrators 65P40: Nonlinear stabilities
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e5
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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