A large probability averaging Theorem for the defocousing NLS

TL;DR

This paper proves that for the defocusing nonlinear Schrödinger equation on a 1D torus, the Fourier coefficients' modulus remains nearly constant over long times for large measure initial data, using Hamiltonian perturbation techniques.

Contribution

It introduces a large probability averaging theorem for the defocusing NLS, extending Hamiltonian perturbation methods to the PDE setting with Gibbs measure initial data.

Findings

Fourier coefficients' modulus remains approximately constant over long times

Results hold for initial data of large measure with respect to Gibbs measure

Theorem applies to times of order ^{2+\u03b6} with  as inverse temperature

Abstract

We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order $\beta^{2+\varsigma}$, $\beta$ being the inverse of the temperature and $\varsigma$ a positive number (we prove $\varsigma= 1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.