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

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Date
2015-05-18MFO Scientific Program
Research in Pairs 2014Series
Oberwolfach Preprints;2015,04Author
Genovese, Giuseppe
Lucatti, Renato
Valeri, Daniele
Metadata
Show full item recordOWP-2015-04
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 $\int h_{2k}, k \geq 1$.