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Multi-Soliton Solutions for the Combined KdV–mKdV Equation in Terms of Wronskian with Multi-Wave and Periodic Cross-Kink Dynamics

Author

Listed:
  • Reem Abdullah Aljethi

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia)

  • Nida Raees

    (Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan)

  • Irfan Mahmood

    (Center for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan)

  • Ejaz Hussain

    (Institute of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan)

Abstract

This article examines the integrability of the combined KdV-mKdV equation, which provides an effective framework for modeling coherent structures in turbulent flows. We generate the explicit Darboux solutions for the combined KdV-mKdV equation using Wronskians. These results are further generalized to the K -th order and supplemented as the logarithmic derivative of the K -th order Wronskian that provides us with the multi-soliton solutions. We generate the exact explicit solution for one-, two-, and three-solitons. Graphical depictions of the soliton formations’ interactions, dynamical characteristics, and temporal evolution are used to support these conclusions. Furthermore, we generate the multi-wave and periodic cross-kink wave solutions by employing bilinear formulism. The graphical representations of these nonlinear excitations highlight their extensive dynamical activity and structural complexity.

Suggested Citation

  • Reem Abdullah Aljethi & Nida Raees & Irfan Mahmood & Ejaz Hussain, 2026. "Multi-Soliton Solutions for the Combined KdV–mKdV Equation in Terms of Wronskian with Multi-Wave and Periodic Cross-Kink Dynamics," Mathematics, MDPI, vol. 14(9), pages 1-16, April.
    • Handle: RePEc:gam:jmathe:v:14:y:2026:i:9:p:1488-:d:1930896
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