A Wong--Zakai resonance-based integrator for nonlinear Schr\"odinger equation with white noise dispersion

Abstract

We introduce a novel approach to numerical approximation of nonlinear Schr\"odinger equation with white noise dispersion in the regime of low-regularity solutions. Approximating such solutions in the stochastic setting is particularly challenging due to randomized frequency interactions and presents a compelling challenge for the construction of tailored schemes. In particular, we design the first resonance-based schemes for this equation, which achieve provable convergence for solutions of much lower regularity than previously required. A crucial ingredient in this construction is the Wong--Zakai approximation of stochastic dispersive system, which introduces piecewise linear phases that capture nonlinear frequency interactions and can subsequently be approximated to construct resonance-based schemes. We prove the well-posedness of the Wong--Zakai approximated equation and establish its proximity to the original full stochastic dispersive system. Based on this approximation, we demonstrate an improved strong convergence rate for our new scheme, which exploits the stochastic nature of the dispersive terms. Finally, we provide numerical experiments underlining the favourable performance of our novel method in practice.