Approximated exponential integrators for the stochastic Manakov equation

Abstract

This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is $ 1/2 $. This is then used to prove that the approximated exponential integrator has convergence order $ 1/2 $ in probability and almost sure order $ 1/2^{-} $, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the approximated exponential integrator as well as a modified version of it.