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Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach

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  • Xu, Guoan
  • Zhang, Yi
  • Li, Jibin

Abstract

Using the bifurcation theory of the planar dynamical system, we study the exact solutions of the complex Ginzburg–Landau equation which is a popular model in mathematical physics. All possible exact explicit parametric representations of traveling wave solutions are given under different parameter conditions, including the solitary wave solutions, periodic wave solutions, compacton solutions pseudo-peakon solutions and periodic peakon solutions. In more general parametric conditions, all possible solutions are caught in one dragnet.

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  • Xu, Guoan & Zhang, Yi & Li, Jibin, 2022. "Exact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 157-167.
    • Handle: RePEc:eee:matcom:v:191:y:2022:i:c:p:157-167
    DOI: 10.1016/j.matcom.2021.08.007
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    References listed on IDEAS

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    1. Djob, Roger Bertin & Kenfact-Jiotsa, Aurelien & Govindarajan, A., 2020. "Non-Lagrangian approach for coupled complex Ginzburg-Landau systems with higher order-dispersion," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    2. Kudryashov, Nikolay A., 2020. "Highly dispersive solitary wave solutions of perturbed nonlinear Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    3. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
    4. Shi, Dongyang & Liu, Qian, 2020. "Superconvergence analysis of a two grid finite element method for Ginzburg–Landau equation," Applied Mathematics and Computation, Elsevier, vol. 365(C).
    5. Kudryashov, Nikolay A., 2020. "First integrals and general solution of the complex Ginzburg-Landau equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    6. Mancas, Stefan C. & Choudhury, S.Roy, 2007. "The complex cubic–quintic Ginzburg–Landau equation: Hopf bifurcations yielding traveling waves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(4), pages 281-291.
    7. Qiu, Yunli & Malomed, Boris A. & Mihalache, Dumitru & Zhu, Xing & Zhang, Li & He, Yingji, 2020. "Soliton dynamics in a fractional complex Ginzburg-Landau model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Zhu, Wenjing & Xia, Yonghui & Bai, Yuzhen, 2020. "Traveling wave solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 382(C).
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