Proof of the kinetic conjecture in a weakly nonlinear Schrodinger equation with random initial data

Jani Lukkarinen
Suomen Akatemia (Academy of Finland)

We report on first progress in rigorous control of a kinetic scaling limit of a weakly nonlinear perturbation of wave-type evolution, here a discrete Schrodinger 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 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. 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].

Presentation (PDF File)

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