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

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.

本文将三次非线性薛定谔方程解释为一个具有无穷维相空间的哈密顿系统。存在一个吉布斯测度，它在与正则运动方程相关的流作用下是不变的。对数索伯列夫不等式和测度集中不等式对吉布斯测度成立，并且在此被推广到k点相关函数以及相关经验测度的分布。根据桥本定理，非线性薛定谔方程给出了一对耦合常微分方程的拉克斯对，其解给出了一个活动标架系统。本文研究了由吉布斯测度在活动标架上诱导的测度的演化；通过数值模拟对结果进行了说明。本文包含了对经验分布在瓦塞斯坦度量下收敛速度的具有良好控制常数的定量估计。