Strong rates of convergence of a splitting scheme for Schr\"{o}dinger equations with nonlocal interaction cubic nonlinearity and white noise dispersion

TL;DR

This paper analyzes a splitting scheme for a stochastic Schrödinger equation with nonlocal cubic nonlinearity, proving strong convergence of order one in mean and demonstrating its norm-preserving property through numerical experiments.

Contribution

It establishes the strong convergence order of a splitting integrator for a stochastic Schrödinger equation with nonlocal nonlinearity and noise, including norm preservation.

Findings

The splitting scheme converges with order one in the p-th mean sense.

The scheme preserves the L^2-norm exactly.

Numerical experiments confirm the theoretical convergence and stability.

Abstract

We analyse a splitting integrator for the time discretization of the Schr\"odinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has order of convergence one in the $p$-th mean sense, for any $p\geq1$ in some Sobolev spaces. We prove that the splitting schemes preserves the $L^2$-norm, which is a crucial property for the proof of the strong convergence result. Finally, numerical experiments illustrate the performance of the proposed numerical scheme.