Vol. 5, No. 5, 2012

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Effective integrable dynamics for a certain nonlinear wave equation

Patrick Gérard and Sandrine Grellier

Vol. 5 (2012), No. 5, 1139–1155
Abstract

We consider the following degenerate half-wave equation on the one-dimensional torus:

itu |D|u = |u|2u,u(0,) = u 0.

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 Hs 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