Low regularity symplectic schemes for stochastic NLS

TL;DR

This paper develops symplectic resonance-based numerical schemes for stochastic nonlinear Schrödinger equations, focusing on convergence analysis and extending deterministic methods to stochastic settings.

Contribution

It introduces a new class of symplectic schemes for stochastic NLS, adapting resonance-based methods from deterministic PDEs to stochastic contexts.

Findings

Derived a resonance-based midpoint rule for stochastic NLS

Analyzed the convergence properties of the proposed scheme

Extended deterministic symplectic schemes to stochastic equations

Abstract

We introduce a class of symplectic resonance based schemes for Schr\"odinger's equation in dimension one, building on the work in [1] wherein resonance based numerical schemes were developed in the context of dispersive PDE driven by time dependent, or space-time dependent, coloured noise. We work primarily with a cubic nonlinearity, advancing the approach introduced in [15] for deriving symplectic schemes in the deterministic setting. As an example of such a scheme we derive the resonance based midpoint rule for the Stochastic NLS and analyse its convergence properties.