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Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon equation

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  • Zhang, Bei
  • Xia, Yonghui
  • Zhu, Wenjing
  • Bai, Yuzhen

Abstract

This paper concerns the qualitative behavior and exact traveling wave solutions of the generalized double combined sinh–cosh-Gordon equation. This equation generalizes sinh-Gordon equation, Liouville equation, Dodd–Bullough–Mikhailov equation, Tzitzeica–Dodd-Bullough equation and Zhiber-Shabat equation as special cases. Thus, this paper presents a unified analysis to find the exact solutions of these known equations. Our results generalize many previous known results (Chen et al., 2009; Fan et al., 2011; Geng et al., 2007; He and Meng, 2017; He et al. 2014; Li and Li 2005; Seadawy et al. 2017; Tang and Huang 2007). We find different kinds of exact solutions such as bright soliton, dark soliton, kink wave, anti-kink wave solutions and periodic wave solutions. Moreover, the explicit expressions of the bounded exact traveling wave solutions are given. Finally, a conclusion ends this paper.

Suggested Citation

  • Zhang, Bei & Xia, Yonghui & Zhu, Wenjing & Bai, Yuzhen, 2019. "Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    • Handle: RePEc:eee:apmaco:v:363:y:2019:i:c:16
    DOI: 10.1016/j.amc.2019.124576
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    References listed on IDEAS

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    1. Li, Jibin & Li, Ming, 2005. "Bounded travelling wave solutions for the (n+1)-dimensional sine- and sinh-Gordon equations," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1037-1047.
    2. Jianming Zhang & Lijun Zhang & Chaudry Masood Khalique, 2014. "Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-7, March.
    3. Wazwaz, Abdul-Majid, 2007. "A variable separated ODE method for solving the triple sine-Gordon and the triple sinh-Gordon equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 703-710.
    4. Yahong Peng & Tonghua Zhang, 2014. "Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, April.
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    Cited by:

    1. 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).
    2. Lewa’ Alzaleq & Du’a Al-zaleq & Suboh Alkhushayni, 2022. "Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients," Mathematics, MDPI, vol. 10(5), pages 1-11, March.
    3. Tang, Lu & Chen, Shanpeng, 2022. "Traveling wave solutions for the diffusive Lotka–Volterra equations with boundary problems," Applied Mathematics and Computation, Elsevier, vol. 413(C).

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