 0912 Jani Lukkarinen, Herbert Spohn
 Weakly nonlinear Schrodinger equation with random initial data
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Jan 21, 09

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Abstract. There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schrodinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution psi_t(x) of the nonlinear Schrodinger equation yields then a stochastic process stationary in x \in Z^d and t \in R. If lambda denotes the strength of the nonlinearity, we prove that the spacetime covariance of psi_t(x) has a limit as lambda > 0 for t=lambda^{2} tau, with tau fixed and tau sufficiently small. The limit agrees with the prediction from kinetic theory.
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