Probability and Pathwise Order of Convergence of a Semidiscrete Scheme for the Stochastic Manakov Equation

Abstract

It is well accepted by physicists that the Manakov PMD equation is a good model to describe the evolution of nonlinear electric fields in optical fibers with randomly varying birefringence. In the regime of the diffusion approximation theory, an effective asymptotic dynamics has recently been obtained to describe this evolution. This equation is called the stochastic Manakov equation. In this article, we propose a semidiscrete version of a Crank--Nicolson scheme for this limit equation and we analyze the approximation error. Allowing sufficient regularity of the initial data, we prove that the numerical scheme has probability order $1/2$ and almost sure order $1/2-\epsilon$ for any $\epsilon \in (0, 1/2)$.