Numerical Method for Stochastic Nonlinear Schrödinger Equation Driven by Multivariate Gaussian Measure: Algorithms and Applications

Abstract

In this paper, we present a novel Galerkin spectral method for numerically solving the stochastic nonlinear Schrödinger (NLS) equation driven by multivariate Gaussian random variables, including the nonlinear term. Our approach involves deriving the tensor product of triple random orthogonal basis and random functions, which enables us to effectively handle the stochasticity and nonlinear term in the equation. We apply this newly proposed method to solve both one- and two-dimensional stochastic NLS equations, providing detailed analysis and comparing the results with Monte Carlo simulation. In addition, the proposed method is applied to the stochastic Ginzburg–Landau equation. Our method exhibits excellent performance in both spatial and random spaces, achieving spectral accuracy in the numerical solutions.