Stochastic nonlinear Schr\"odinger equations on tori

TL;DR

This paper studies stochastic nonlinear Schrödinger equations on tori, establishing well-posedness results for various nonlinearities and stochastic forcing types, advancing understanding of their mathematical properties.

Contribution

It proves global well-posedness for 1D cubic SNLS and local/global results for other nonlinearities, extending the theory of SNLS on tori.

Findings

Global well-posedness for 1D cubic SNLS in L^2

Local well-posedness for superquintic and supercubic cases

Global well-posedness in energy space for defocusing energy-subcritical problems

Abstract

We consider the stochastic nonlinear Schr\"odinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.