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New families of soliton solutions and dynamics of nonlinear traveling waves for the Whitham–Broer–Kaup equation

Author

Listed:
  • Pradhan, Dickcha
  • Ali, Khalid K.
  • Karakoc, Seydi Battal Gazi
  • Saha, Asit

Abstract

In this article, we consider the Whitham–Broer–Kaup (WBK) equations, which are a significant shallow water equation, as it can be useful in describing the dispersive long waves in shallow water. Our study consists of two main parts: in the first part, we propose the general form of the Kudryashov method to create some different and new exact traveling wave solutions of the equation. Also in this part, the behavior of the exact solutions are represented graphically. The Kudryashov method transforms the WBK equations into a set of algebraic equations, which are solved to obtain the analytical solutions. This method can also be applied to both integrable and non-integrable equations. The method is a dominant approach for generating exact solutions of nonlinear evolution equations. In the second part, dynamics of nonlinear traveling waves for the WBK equations are studied in the presence of external periodic perturbation. A three-dimensional dynamical system is obtained by using a traveling wave transformation in the presence of an external periodic perturbation from the nonlinear WBK equations. Multiperiodic, chaotic and quasiperiodic features are shown through time-series plots and phase-projection plots by varying the parameters w and g0.

Suggested Citation

  • Pradhan, Dickcha & Ali, Khalid K. & Karakoc, Seydi Battal Gazi & Saha, Asit, 2026. "New families of soliton solutions and dynamics of nonlinear traveling waves for the Whitham–Broer–Kaup equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 239(C), pages 376-390.
    • Handle: RePEc:eee:matcom:v:239:y:2026:i:c:p:376-390
    DOI: 10.1016/j.matcom.2025.05.016
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    References listed on IDEAS

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    1. Khalid K. Ali & M. S. Mehanna & M. Ali Akbar & Prasun Chakrabarti & Kenan Yildirim, 2022. "Analytical Soliton Solutions of the Coupled Radhakrishnan-Kundu-Lakshmanan Equation via Three Techniques," Journal of Mathematics, Hindawi, vol. 2022, pages 1-13, October.
    2. Ming Song, 2012. "Application of Bifurcation Method to the Generalized Zakharov Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-8, November.
    3. M. M. Rashidi & D. D. Ganji & S. Dinarvand, 2008. "Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method," International Journal of Differential Equations, Hindawi, vol. 2008, pages 1-8, May.
    4. Ming Song, 2012. "Application of Bifurcation Method to the Generalized Zakharov Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    5. G.C. Layek, 2015. "An Introduction to Dynamical Systems and Chaos," Springer Books, Springer, number 978-81-322-2556-0, January.
    6. Chen, Aihua & Li, Xuemei, 2006. "Darboux transformation and soliton solutions for Boussinesq–Burgers equation," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 43-49.
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