Energy cascades for NLS on the torus

Rémi Carles; Erwan Faou

doi:10.3934/dcds.2012.32.2063

Article Contents

2012, Volume 32, Issue 6: 2063-2077. Doi: 10.3934/dcds.2012.32.2063

This issue Previous Article Absolute and delay-dependent stability of equations with a distributed delay Next Article A minimal approach to the theory of global attractors

Energy cascades for NLS on the torus

Rémi Carles^1, and
Erwan Faou^2,

1.
CNRS & Univ. Montpellier 2, UMR 5149, CC 051, F-34095 Montpellier
2.
INRIA & ENS Cachan Bretagne, Avenue Robert Schumann, F-35170 Bruz

Received: March 31, 2011

Revised: June 30, 2011

Published: May 2012

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Abstract

We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35C20, 37K55.

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