Quasi-invariant Gaussian measures for the cubic nonlinear Schr\"odinger equation with third order dispersion

TL;DR

This paper proves that certain Gaussian measures are quasi-invariant under the flow of a cubic nonlinear Schrödinger equation with third order dispersion on the circle, using gauge transformations to handle resonances.

Contribution

It establishes quasi-invariance of Gaussian measures for the equation in the non-resonant case, extending understanding of measure behavior in dispersive PDEs.

Findings

Gaussian measures are quasi-invariant for the equation

Gauge transformations effectively remove resonant dynamics

Results apply to Sobolev spaces with s > 3/4

Abstract

In this paper, we consider the cubic nonlinear Schr\"odinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac 34$, are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.