Lie-Trotter splitting for the nonlinear stochastic Manakov system. (English) Zbl 1503.65007
Summary: 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 \(\frac{1}{2}-\) 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.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65J08 | Numerical solutions to abstract evolution equations |
Keywords:
stochastic partial differential equations; stochastic Manakov equation; coupled system of stochastic nonlinear Schrödinger equations; numerical schemes; splitting scheme; Lie-Trotter scheme; strong convergence; convergence in probability; almost sure convergence; convergence rates; blowupReferences:
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