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The complex cubic–quintic Ginzburg–Landau equation: Hopf bifurcations yielding traveling waves

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  • Mancas, Stefan C.
  • Choudhury, S.Roy

Abstract

In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic–quintic Ginzburg–Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:74:y:2007:i:4:p:281-291
    DOI: 10.1016/j.matcom.2006.10.022
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    Cited by:

    1. 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.
    2. Mancas, Stefan C. & Adams, Ronald, 2019. "Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 385-393.

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