Stochastic Symplectic and Multi-Symplectic Methods for Nonlinear Schr\"odinger Equation with White Noise Dispersion

TL;DR

This paper introduces stochastic symplectic and multi-symplectic numerical methods for the nonlinear Schrödinger equation with white noise dispersion, preserving key physical invariants and demonstrating convergence.

Contribution

It develops novel structure-preserving numerical schemes for stochastic nonlinear Schrödinger equations, ensuring charge conservation and proving their convergence properties.

Findings

Methods preserve continuous and discrete charge conservation laws.

Proposed schemes are convergent with temporal order one in probability.

Numerical experiments confirm theoretical properties.

Abstract

We indicate that the nonlinear Schr\"odinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.