Title: Decay of equilibrium time correlations in a weakly nonlinear
Schroedinger equation
Abstract:
We report on a first progress in rigorous control of the kinetic
scaling limit of a weakly nonlinear perturbation of wave-type
evolution, here a discrete Schroedinger equation. Since we consider 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 of the nonlinear Schroedinger equation yields then a
stochastic process stationary in space and time. If lambda denotes
the strength of the nonlinearity, we prove that the space-time
covariance of the field 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. The talk is
based on a joint work with Herbert Spohn [J. Lukkarinen and H. Spohn,
Weakly nonlinear Schroedinger equation with random initial data,
preprint arXiv:0901.3283].