Vol. 5, No. 5, 2012

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
 
Author index
To appear
 
Other MSP journals
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