Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation

Author:

Blower Gordon1ORCID,Khaleghi Azadeh2ORCID,Kuchemann-Scales Moe1ORCID

Affiliation:

1. Department of Mathematics and Statistics, Lancaster University 1 , Lancaster, United Kingdom

2. ENSAE–CREST, Institut Polytechnique de Paris 2 , Palaiseau, France

Abstract

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

Publisher

AIP Publishing

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