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Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

Author

Listed:
  • Lewa’ Alzaleq

    (Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan)

  • Du’a Al-zaleq

    (Computer Information Science Department, Minnesota State University, Mankato, MN 56001, USA)

  • Suboh Alkhushayni

    (Computer Information Science Department, Minnesota State University, Mankato, MN 56001, USA)

Abstract

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:822-:d:764166
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    References listed on IDEAS

    as
    1. Wazwaz, Abdul-Majid, 2006. "Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 127-135.
    2. Gabriel Magalakwe & Chaudry Masood Khalique, 2013. "New Exact Solutions for a Generalized Double Sinh-Gordon Equation," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-5, August.
    3. Grauel, A., 1985. "Sinh-Gordon equation, Painlevé property and Bäcklund transformation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 132(2), pages 557-568.
    4. 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.
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