A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers

Abstract

In this article we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schrödinger equations. This system arises in the study of pulse propagation in randomly birefringent optical fibers. We first show existence and uniqueness of solutions for the random PDE and the limiting equation. We follow the work of Garnier and Marty [Wave Motion 43 (2006) 544-560], Marty [Problèmes d'évolution en milieux aléatoires: Théorèmes limites, schémas numériques et applications en optique (2005) Univ. Paul Sabatier], where a linear electric field is considered, and we get an asymptotic dynamic for the nonlinear electric field.

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

“…We now recall some results obtained in [6,13] on the existence of a solution for the system (1.1). Let (Ω, F, P) be a probability space on which is defined a 3-dimensional Brownian motion W (t) = (W k (t)) k=1,2,3 .…”

“…In section 1.2, we introduce some notations and the main result of this article. Then, following the approach of [6,13] for the continuous equation, we construct a discrete random propagator associated to the linear equation. In section 2, we study the linear Euler scheme with semi-implicit discretization of the noise and prove that the strong order is 1/2.…”

“…To prove this result, we will use the next Lemma whose proof relies on the same arguments as in [3,6].…”

“…In our case, the difficult and innovative point lies in the linear estimate. Indeed, the noise term contains a one order derivative and hence cannot be treated as a perturbation [6,13]. Moreover an implicit discretization of the noise has to be considered to build a conservative scheme and the delicate point, in order to obtain the strong error, is to deal with random matrices.…”

“…In optics, spectral methods are very often used to solve the nonlinear Schrödinger equation because the group associated to the free equation has an explicit and very simple form. The random propagator, solution of the linear equation associated to (1.1), does not have an explicit formulation in Fourier space[6,13]. Consequently a numerical approximation of the linear equation is obtained resolving a linear system.…”