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This article proposes an advanced algorithm for the numerical solution of a coupled nonlinear Schrödinger equations system describing the evolution of a high-power femtosecond optical pulse in a single-mode polarization-maintaining optical fiber. We use the algorithm based on a variant of the split-step method with the Madelung transform to calculate the complex amplitude when executing a nonlinear operator. In contrast to the known solution, the proposed algorithm eliminates the need to numerically solve differential equations directly, concerning the phase of complex amplitude when executing the nonlinear operator. This made it possible, other things being equal, to reduce the computation time by more than four times. [ABSTRACT FROM AUTHOR]

The authors wish to make a change to the author names (adding a new author - Ildaris M. Gabdulkhakov) on this paper [[1]]. On page 10, the Funding sources are updated as follows: B Funding: b A.Z.S. was funded by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-03-2020-051, Topic No. fzsu-2020-0020); I.M.G. was funded by the Russian Foundation for Basic Research: 19-37-90057; A.A.K. was funded by a grant from the President of the Russian Federation for state support of young Russian scientists MK-3421.2019.8:075-15-2019-309; O.G.M., V.A.B., and A.V.B. were funded by the Russian Foundation for Basic Research, DST, NSFC, and NRF: 19-57-80016 BRICS t; D.V.I., V.A.I., M.I.R, and V.V.O. were funded by the Russian Science Foundation: 18-19-00401. On page 1, the author affiliation 1 is updated as follows: SP 1 sp Department of Radiophotonics and Microwave Technologies, Kazan National Research Technical University named after A.N. Tupolev-KAI, 10, Karl Marx Street, Kazan, 420111 Rep. Tatarstan, Russia; VIAnfinogentov@kai.ru (V.I.A.); microoil@mail.ru (O.G.M.); AAKuznetsov@kai.ru (A.A.K.); baikerboy@mail.ru (I.M.G.) On page 10, the Author Contributions are updated as follows: B Author Contributions: b Conceptualization, A.Z.S. and D.V.I.; Formal analysis, V.I.A., D.V.I., V.A.I., and V.A.B.; Funding acquisition, O.G.M., A.V.B., M.I.R., A.A.K., and I.M.G.; Investigation, A.Z.S., V.I.A., V.A.B., A.A.K., D.V.I., V.A.I., M.I.R., and I.M.G.; Methodology, A.Z.S., V.I.A., O.G.M., and V.A.B.; Software, A.Z.S. and V.I.A.; Supervision, V.A.B., and A.V.B.; Validation, A.Z.S., V.I.A., O.G.M., and V.A.B.; Visualization, V.V.O. and A.Z.S.; Writing - review & editing, A.Z.S. All authors have read and agreed to the published version of the manuscript. [Extracted from the article]

This paper discusses novel approaches to the numerical integration of the coupled nonlinear Schrödinger equations system for few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes' interactions. In this case, the coupled nonlinear Schrödinger equation system (CNSES) solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails due to the non-stability of the computing scheme. Owing to this, here we use the combined explicit/implicit finite-difference integration scheme, which is based on the implicit Crank–Nicolson finite-difference scheme. It ensures the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has a higher performance (resolution)—up to three times or more in comparison with the split-step Fourier method—since there is no need to produce direct and inverse Fourier transforms at each integration step. The key advantage of the developed approach is the calculation of any number of modes propagated in the fiber. [ABSTRACT FROM AUTHOR]

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system, different from the generally accepted approach based on the method of splitting according to physical processes. A combined explicit/implicit finite-difference integration scheme based on the implicit Crank–Nicolson finite-difference scheme is proposed and substantiated. It allows the integration of a nonlinear system of equations with a choice of nonlinear terms from the previous integration step. The main advantages of the proposed method are: its absolute stability through the use of an implicit finite-difference integration scheme and an integrated mechanism for refining the numerical solution at each step; integration with automatic step selection; performance gains (or resolutions) up to three or more orders of magnitude due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step, as is required in the method of splitting according to physical processes. An additional advantage of the proposed method is the ability to calculate the interaction with an arbitrary number of propagation modes in the fiber. [ABSTRACT FROM AUTHOR]