Low regularity symplectic schemes for stochastic NLS

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.