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Lie–Trotter Splitting for the Nonlinear Stochastic Manakov System

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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 \({{\frac{1}{2}-}}\) 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.

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All data generated or analysed during this study are included in this published article.

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Acknowledgements

We appreciate the referees’ comments on an earlier version of the paper. Part of the work of D. Cohen was carried out when working for the Department of Mathematics and Mathematical Statistics at Umeå University.

Funding

This work was partially supported by the Swedish Research Council (VR) (project nr. \(2018-04443\)), FRÖ the mobility programs of the French Embassy/Institut français de Suède, and INRIA Lille Nord-Europe. G. Dujardin was partially supported by the Labex CEMPI (ANR-11-LABX-0007-01). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N, Umeå University.

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Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187, Umeå, Sweden

    André Berg

  2. Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-41296, Gothenburg, Sweden

    David Cohen

  3. Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000, Villeneuve-d’Ascq, France

    Guillaume Dujardin

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  1. André Berg
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  2. David Cohen
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  3. Guillaume Dujardin
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Correspondence to Guillaume Dujardin.

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Berg, A., Cohen, D. & Dujardin, G. Lie–Trotter Splitting for the Nonlinear Stochastic Manakov System. J Sci Comput 88, 6 (2021). https://doi.org/10.1007/s10915-021-01514-y

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  • DOI: https://doi.org/10.1007/s10915-021-01514-y

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