Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

Abstract

We study the one dimensional periodic derivative nonlinear Schrödinger (DNLS) equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion $\int h_k, k \in \mathbb{Z}_+$. In each $\int h_{2k}$ the term with the highest regularity involves the Sobolev norm $\dot{H}^k(\mathbb{T})$ of the solution of the DNLS equation. We show that a functional measure on $L^2(\mathbb{T})$, absolutely continuous w.r.t. the Gaussian measure with covariance $(\mathbb{I}+(-\Delta)^k)^{-1}$, is associated to each integral of motion $\int h_{2k}, k \geq 1$.