The wave turbulence theory describes the nonequilibrium statistical mechanics for a large class of nonlinear dispersive systems. A major goal of this theory is to derive the wave kinetic equation, which predicts the behavior of macroscopic limits of ensemble averages for microscopic interacting systems. This occurs at a particular "kinetic time scale" in the "weak-nonlinearity" limit where the number of interacting modes goes to infinity and the nonlinearity strength goes to zero. For nonlinear Schrodinger equations such limits have been derived on a formal level and studied extensively since the 1920s, but a rigorous proof remains open.

We provide the first rigorous derivation of wave kinetic equation, which reaches the kinetic time scale less an arbitrary small power, in a particular scaling regime for the number of modes and nonlinearity strength. We expect that our method can also be extended to other equations; in the end we make a short discussion about water waves. This is joint work with Zaher Hani.