#### 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:

$i{\partial }_{t}u-|D|u=|u{|}^{2}u,\phantom{\rule{1em}{0ex}}u\left(0,\cdot \phantom{\rule{0.3em}{0ex}}\right)={u}_{0}.$

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 ${H}^{s}$ 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