On a splitting scheme for the nonlinear Schrödinger equation in a random medium

Abstract. In this paper we consider a nonlinear Schrödinger equation (NLS) with random coefficients, in a regime of separation of scales corresponding to diffusion approximation. The primary goal of this paper is to propose and study an efficient numerical scheme in this framework. We use a pseudo-spectral splitting scheme and we establish the order of the global error. In particular we show that we can take an integration step larger than the smallest scale of the problem, here the correlation length of the random medium. We study the asymptotic behavior of the numerical solution in the diffusion approximation regime. Key words. Light waves, random media, asymptotic theory, splitting sheme. AMS subject classifications. 35Q55, 60F05, 65M12 1. Introduction. Optica