Effective integrable dynamics for some nonlinear wave equation

Patrick Gerard (1) , Sandrine Grellier (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We consider the following degenerate half wave equation on the one dimensional torus $$\quad i\partial _t u-|D|u=|u|^2u, \; u(0,\cdot)=u_0. $$ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system-- the cubic Szegö equation. As a consequence, we prove an instability result for large $H^s$ norms of solutions of this wave equation.

Mots clés: Birkhoff normal form; nonlinear wave equation; perturbation of integrable systems

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Variables complexes [math.CV]

Patrick Gerard, Sandrine Grellier. Effective integrable dynamics for some nonlinear wave equation. Analysis & PDE, 2012, 5-5, pp.1139-1155. ⟨hal-00635686⟩