Hasimoto frames and the Gibbs measure of periodic nonlinear Schr\"odinger Equation

TL;DR

This paper explores the Hamiltonian structure of the periodic nonlinear Schrödinger equation, analyzing the invariance and concentration properties of the Gibbs measure and its evolution under Hasimoto's framework.

Contribution

It extends the understanding of Gibbs measures for NLSE, linking them with Hasimoto frames and analyzing their measure-theoretic properties and evolution.

Findings

Gibbs measure invariance under NLSE flow

Logarithmic Sobolev inequalities hold for the Gibbs measure

Evolution of measures on moving frames studied

Abstract

The paper interprets the cubic nonlinear Schr\"odinger equation as a Hamiltonian system with infinite dimensional phase space. There is 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, NLSE gives a Lax pair of coupled ODE 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.