Iterative Splitting Methods: Almost Asymptotic Symplectic Integrator for Stochastic Nonlinear Schr\"odinger Equation

TL;DR

This paper introduces iterative splitting methods for stochastic nonlinear Schrödinger equations that nearly preserve symplectic structure, combining analytical decoupling with accelerated numerical schemes.

Contribution

It develops almost symplectic integrators based on iterative splitting and contraction mapping principles for stochastic nonlinear Schrödinger equations.

Findings

Methods nearly conserve symplectic structure.

Decoupling stochastic equations accelerates computations.

Numerical experiments validate the approach.

Abstract

In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is based on rewriting an iterative splitting approach as a successive approximation method based on a contraction mapping principle and that we have an almost symplectic scheme. We apply a stochastic differential equation, that we can decouple into a deterministic and stochatic part, while each part can be solved analytically. Such decompositions allow accelerating the methods and preserving, under suitable conditions, the symplecticity of the schemes. A numerical analysis and application to the stochastic Schroedinger equation are presented.