Weakly nonlinear Schr\"odinger equation with random initial data

TL;DR

This paper investigates the validity of the kinetic limit for a discretized weakly nonlinear Schrödinger equation with random initial data, showing convergence of the covariance to kinetic theory predictions as nonlinearity strength diminishes.

Contribution

It provides the first progress towards proving the kinetic limit for a Hamiltonian lattice Schrödinger equation with Gibbs measure initial data.

Findings

Space-time covariance converges as nonlinearity strength approaches zero.

The limiting behavior matches kinetic theory predictions.

Results hold for small fixed times in the weak nonlinearity regime.

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 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 the corresponding 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 space-time 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.