Weakly nonlinear Schrödinger equation with random initial data

Abstract

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈ℤ d and t∈ℝ. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ t (x) has a limit as λ→0 for t=λ −2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.