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Consider the cubic nonlinear Schr\"odinger equation set on a d-dimensionaltorus, with data whose Fourier coefficients have phases which are uniformlydistributed and independent. We show that, on average, the evolution of themoduli of the Fourier coefficients is governed by the so-called wave kineticequation, predicted in wave turbulence theory, on a nontrivial timescale.

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr\"odinger equation