Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation

TL;DR

This paper constructs measures related to the integrals of motion of the periodic derivative nonlinear Schrödinger equation, linking the system's integrability with probabilistic measures on function spaces.

Contribution

It establishes a connection between the integrals of motion of the DNLS equation and specific functional measures on Sobolev spaces, advancing understanding of its probabilistic structure.

Findings

Measures are absolutely continuous with respect to Gaussian measures.

Each measure corresponds to an integral of motion involving Sobolev norms.

The approach links integrability with probabilistic analysis of the DNLS.

Abstract

We study the one dimensional periodic derivative nonlinear Schr\"odinger (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 $\int h_{2k}$, $k\geq1$.