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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Vyacheslav Trofimov & Maria Loginova & Mikhail Fedotov & Daniil Tikhvinskii & Yongqiang Yang & Boyuan Zheng, 2022. "Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation," Mathematics, MDPI, vol. 10(11), pages 1-24, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1912-:d:830927
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    References listed on IDEAS

    as
    1. Barletti, L. & Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2018. "Energy-conserving methods for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 3-18.
    2. Trofimov, Vyacheslav A. & Stepanenko, Svetlana & Razgulin, Alexander, 2021. "Conservation laws of femtosecond pulse propagation described by generalized nonlinear Schrödinger equation with cubic nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 366-396.
    3. Ismail, M.S. & Taha, Thiab R., 2007. "A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(4), pages 302-311.
    4. Wang, Tingchun & Jiang, Jiaping & Wang, Hanquan & Xu, Weiwei, 2018. "An efficient and conservative compact finite difference scheme for the coupled Gross–Pitaevskii equations describing spin-1 Bose–Einstein condensate," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 164-181.
    5. Eduardo Salete & Antonio M. Vargas & Ángel García & Mihaela Negreanu & Juan J. Benito & Francisco Ureña, 2020. "Complex Ginzburg–Landau Equation with Generalized Finite Differences," Mathematics, MDPI, vol. 8(12), pages 1-13, December.
    6. Vyacheslav Trofimov & Maria Loginova, 2021. "Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants," Mathematics, MDPI, vol. 9(21), pages 1-26, October.
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