Derivation of the wave kinetic equation

Abstract:

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. Usually this limit happens at a particular “kinetic time scale” in the “weak-nonlinearity” limit where the number of interacting modes goes to infinity while 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.

In this work, joint with Zaher Hani, we provide the first rigorous derivation of wave kinetic equation, which reaches the kinetic time scale up to an arbitrary small power, in a particular scaling regime for the number of modes and the strength of nonlinearity. We rely on a robust method, which can be extended to other semilinear models, and possibly also to quasilinear models (such as water waves).