Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

Vyacheslav Trofimov, Maria Loginova, Mikhail Fedotov, Daniil Tikhvinskii, Yongqiang Yang and Boyuan Zheng
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Vyacheslav Trofimov: School of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, China
Maria Loginova: The Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, Russia
Mikhail Fedotov: The Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, Russia
Daniil Tikhvinskii: The Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, Russia
Yongqiang Yang: School of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, China
Boyuan Zheng: School of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, China

Mathematics, 2022, vol. 10, issue 11, 1-24

Abstract: In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results.

Keywords: Ginzburg–Landau equation; nonlinear and linear propagations of an optical pulse; conservation laws; finite-difference scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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