Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation
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...
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MDPI AG
2022-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/11/1912 |
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| author | Vyacheslav Trofimov Maria Loginova Mikhail Fedotov Daniil Tikhvinskii Yongqiang Yang Boyuan Zheng |
| author_facet | Vyacheslav Trofimov Maria Loginova Mikhail Fedotov Daniil Tikhvinskii Yongqiang Yang Boyuan Zheng |
| author_sort | Vyacheslav Trofimov |
| collection | DOAJ |
| description | 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. |
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| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:05:32Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-8616f39cdfa64267a48a804804a0774e2023-11-23T14:26:51ZengMDPI AGMathematics2227-73902022-06-011011191210.3390/math10111912Conservative Finite-Difference Scheme for 1D Ginzburg–Landau EquationVyacheslav Trofimov0Maria Loginova1Mikhail Fedotov2Daniil Tikhvinskii3Yongqiang Yang4Boyuan Zheng5School of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, ChinaThe Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, RussiaThe Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, RussiaThe Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University GSP-1, Leninskie Gory, 119991 Moscow, RussiaSchool of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, ChinaSchool of Mechanical and Automotive Engineering, South China University of Technology 381, Wushan Road, Tianhe District, Guangzhou 510641, ChinaIn 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.https://www.mdpi.com/2227-7390/10/11/1912Ginzburg–Landau equationnonlinear and linear propagations of an optical pulseconservation lawsfinite-difference scheme |
| spellingShingle | Vyacheslav Trofimov Maria Loginova Mikhail Fedotov Daniil Tikhvinskii Yongqiang Yang Boyuan Zheng Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation Mathematics Ginzburg–Landau equation nonlinear and linear propagations of an optical pulse conservation laws finite-difference scheme |
| title | Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation |
| title_full | Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation |
| title_fullStr | Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation |
| title_full_unstemmed | Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation |
| title_short | Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation |
| title_sort | conservative finite difference scheme for 1d ginzburg landau equation |
| topic | Ginzburg–Landau equation nonlinear and linear propagations of an optical pulse conservation laws finite-difference scheme |
| url | https://www.mdpi.com/2227-7390/10/11/1912 |
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