Long-term error analysis of low-regularity integrators for stochastic Schr$\ddot{\rm o}$dinger equations (2410.22201v2)

Abstract: In this paper, we design an explicit non-resonant low-regularity integrator for the cubic nonlinear stochastic Schr$\ddot{\rm o}$dinger equation (SNLSE) with the aim of allowing long time simulations. First, we carry out a strong error analysis for the new integrator. Next we provide, for small initial data of size $\mathcal{O}(\varepsilon), \varepsilon \in (0,1]$, and noise of size $\varepsilon^q, \ q>2$, long-term error estimates in the space $L^{2p}(\Omega, H^1), p\ge 1$, revealing an error of size $\mathcal{O}\left(\tau^{\frac{1}{2}}\cdot \varepsilon^{{\rm min}(2,(q-2))}\right)$ up to time $\mathcal{O}(\varepsilon^{-2})$. This is achieved with the regularity compensation oscillation technique \cite{sch0,bao}, which has been here introduced and exploited for the stochastic setting. A numerical experiment confirms the superior long-time behaviour of our new scheme, compared to other existing integrators.