Effective integrable dynamics for a certain nonlinear wave equation

We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system—the cubic Szegő equation. As a consequence, we prove an instability result for large $H^{s}$ norms of solutions of this wave equation.

Keywords

Birkhoff normal form, nonlinear wave equation, perturbation of integrable systems

Mathematical Subject Classification 2010

Primary: 35B34, 35B40, 37K55

Milestones

Received: 26 October 2011

Revised: 1 June 2012

Accepted: 6 August 2012

Published: 29 December 2012

Authors

Patrick Gérard
Laboratoire de Mathématiques d’Orsay CNRS, UMR 8628 Université Paris-Sud XI 91405 Orsay France

Sandrine Grellier
Département de Mathématiques MAPMO-UMR 6628 Université Orléans 45047 Orleans Cedex 2 France

Vol. 5, No. 5, 2012

Patrick Gérard and Sandrine Grellier

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors