Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

Consider the cubic nonlinear Schrödinger equation set on a d -dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation , predicted in wave turbulence theory, on a nontrivial timescale.