Effective integrable dynamics for a certain nonlinear wave equation

Abstract

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

$$i{\partial }_{t}u-\left|D\right|u=|u{|}^{2}u,u\left(0,\cdot \right)=u_0.$$

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.