Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System

TL;DR

This paper studies the convergence properties of the Lie-Trotter splitting scheme applied to the stochastic Manakov equation, demonstrating convergence orders and numerical behavior for different nonlinearities in optical fiber models.

Contribution

It provides the first rigorous analysis of convergence order for the Lie-Trotter scheme on the stochastic Manakov system, including numerical validation and blowup investigation.

Findings

Strong convergence order is 1/2 for Lipschitz nonlinearities.

Convergence order is 1/2 in probability and almost sure for cubic nonlinearities.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.