Effective integrable dynamics for some nonlinear wave equation

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