Effective integrable dynamics for some nonlinear wave equation

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