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Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

Author

Listed:
  • Aly R. Seadawy

    (Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia)

  • Syed T. R. Rizvi

    (Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Punjab 54000, Pakistan)

  • Hanadi Zahed

    (Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah 41411, Saudi Arabia)

Abstract

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M -shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

Suggested Citation

  • Aly R. Seadawy & Syed T. R. Rizvi & Hanadi Zahed, 2023. "Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficien," Mathematics, MDPI, vol. 11(5), pages 1-16, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1074-:d:1075376
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    References listed on IDEAS

    as
    1. He, Ji-Huan & Wu, Xu-Hong, 2006. "Exp-function method for nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 700-708.
    2. Rizvi, Syed T.R. & Seadawy, Aly R. & Ahmed, Sarfaraz & Younis, Muhammad & Ali, Kashif, 2021. "Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Seadawy, Aly R. & Bilal, M. & Younis, M. & Rizvi, S.T.R. & Althobaiti, Saad & Makhlouf, M.M., 2021. "Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
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    5. Abourabia, A.M. & El Horbaty, M.M., 2006. "On solitary wave solutions for the two-dimensional nonlinear modified Kortweg–de Vries–Burger equation," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 354-364.
    6. Seadawy, Aly R. & Rizvi, Syed T.R. & Mustafa, B. & Ali, K. & Althubiti, Saeed, 2022. "Chirped periodic waves for an cubic-quintic nonlinear Schrödinger equation with self steepening and higher order nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    7. Rizvi, Syed T.R. & Seadawy, Aly R. & Farah, N. & Ahmad, S., 2022. "Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
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