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