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A Comprehensive Study of Dynamical Behavior and Nonlinear Structures of the Modified α Equation

Author

Listed:
  • Hassan Almusawa

    (Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia)

  • Musawa Yahya Almusawa

    (Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia)

  • Adil Jhangeer

    (IT4Innovations, VSB—Technical University of Ostrava, 70800 Ostrava-Poruba, Czech Republic)

  • Zamir Hussain

    (Department of Mathematics, University of Wah, Wah Cantt 47040, Pakistan)

Abstract

In this article, the modified α equation is solved using the direct algebraic approach. As a result, numerous new and more generalized exact solutions for such equations have been found, taking into account the wide range of travelling structures. The rational, trigonometric, hyperbolic, and exponential functions with a couple of licentious parameters are thus included in these exact answers. Analytical solutions feature a variety of physical structures, which are visually studied to demonstrate their dynamic behavior in 2D and 3D. Considering the parameters, all feasible phase portraits are shown. Furthermore, we used numerical approaches to determine the nonlinear periodic structures of the mentioned model, and the data are graphically displayed. Additionally, we employed numerical approaches to determine the nonlinear conditions that contribute to the presented model, and the data are graphically displayed. After evaluating the influence of frequency following the application of an external periodic factor, sensitivity exploration is used to study quasi-periodic and chaotic behavior for several starting value problems. Furthermore, the function of physical characteristics is investigated using an external periodic force. Quasi-periodic and quasi-periodic-chaotic patterns are described with the inclusion of a perturbation term. The direct algebraic methodology would be used to derive the soliton solution of modified α equation, from which the Galilean transformation derives traveling wave solutions of the considered and a bifurcation behavior is reported. Analytical and numerical methods have been used to have the condition of the travelling wave phase transformation. The well-judged values of parameters are enhanced well with a graphically formal analysis of such specific solutions to illustrate their propagation. Then a planer dynamical system is introduced, and a bifurcation analysis is utilized to identify the bifurcation structures of the dynamical model’s nonlinear wave propagation solutions. Additionally, the periodic and quasi-periodic behavior of the discussed equation is analyzed using sensitivity analysis for a range of beginning values. To further comprehend the dynamical behaviors of the resultant solutions, a graphic analysis is conducted.

Suggested Citation

  • Hassan Almusawa & Musawa Yahya Almusawa & Adil Jhangeer & Zamir Hussain, 2024. "A Comprehensive Study of Dynamical Behavior and Nonlinear Structures of the Modified α Equation," Mathematics, MDPI, vol. 12(23), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3809-:d:1534712
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    References listed on IDEAS

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    1. Natiq, Hayder & Banerjee, Santo & Misra, A.P. & Said, M.R.M., 2019. "Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 58-68.
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