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arXiv · 1502.05967

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

Giuseppe Genovese·Renato Lucà·Daniele Valeri
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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 $\int h_{2k}$, $k\geq1$.

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Giuseppe Genovese, Renato Lucà, Daniele Valeri. 2015-02-20. Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation. https://arxiv.org/abs/1502.05967

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