Effective integrable dynamics for a certain nonlinear wave equation

We consider the following degenerate half-wave equation on the one-dimensional torus: ¶∂−||=||2, (0, ⋅) =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 math norms of solutions of this wave equation.