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hal-00505201, version 1

On the energy exchange between resonant modes in nonlinear Schrödinger equations

Benoit Grebert () 1, Carlos Villegas-Blas () 2

Abstract: We consider the nonlinear Schrödinger equation $$ i\psi_t= -\psi_{xx}\pm 2\cos 2x \ |\psi|^2\psi,\quad x\in S^1,\ t\in \R$$ and we prove that the solution of this equation, with small initial datum $\psi_0=\e (\cos x+\sin x)$, will periodically exchange energy between the Fourier modes $e^{ix}$ and $e^{-ix}$. This beating effect is described up to time of order $\e^{-9/4}$ while the frequency is of order $\e^2$. We also discuss some generalizations.

  • 1:  Laboratoire de Mathématiques Jean Leray (LMJL)
  • CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes
  • 2:  Insituto de Matematicas, Unidad Cuernavaca
  • Universidad Autonoma de Mexico
  • Domain : Mathematics/Dynamical Systems
    Mathematics/Analysis of PDEs
  • Keywords : Normal form – Nonlinear Schrödinger equation – Resonances – Beating effect
 
  • hal-00505201, version 1
  • http://hal.archives-ouvertes.fr/hal-00505201
  • oai:hal.archives-ouvertes.fr:hal-00505201
  • From: Benoit Grebert <>
  • Submitted on: Friday, 23 July 2010 03:24:00
  • Updated on: Saturday, 24 July 2010 15:00:26