Effective integrable dynamics for a certain nonlinear wave equation

We consider the following degenerate half-wave equation on the one-dimensional torus:¶∂ν−||ν=|ν|2ν,<mspace class="quad" width="1em"/>ν(0,⋅<mspace class="thinspace" width="0.3em"/>)=ν0.¶ 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 [math] norms of solutions of this wave equation.