IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i5p841-d1604470.html
   My bibliography  Save this article

Bifurcation, Quasi-Periodic, Chaotic Pattern, and Soliton Solutions to Dual-Mode Gardner Equation

Author

Listed:
  • Adel Elmandouh

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia)

Abstract

This study aims to investigate various dynamical aspects of the dual-mode Gardner equation derived from an ideal fluid model. By applying a specific wave transformation, the model is reduced to a planar dynamical system, which corresponds to a conservative Hamiltonian system with one degree of freedom. Using Hamiltonian concepts, phase portraits are introduced and briefly discussed. Additionally, the conditions for the existence of periodic, super-periodic, and solitary solutions are summarized in tabular form. These solutions are explicitly constructed, with some graphically represented through their 2 D and 3 D profiles. Furthermore, the influence of specific physical parameters on these solutions is analyzed, highlighting their effects on amplitude and width. By introducing a more general periodic external influence into the model, quasi-periodic and chaotic behavior are explored. This is achieved through the presentation of 2 D and 3 D phase portraits, along with time-series analyses. To further examine chaotic patterns, the Poincaré surface of section and sensitivity analysis are employed. Numerical simulations reveal that variations in frequency and amplitude significantly alter the dynamical characteristics of the system.

Suggested Citation

  • Adel Elmandouh, 2025. "Bifurcation, Quasi-Periodic, Chaotic Pattern, and Soliton Solutions to Dual-Mode Gardner Equation," Mathematics, MDPI, vol. 13(5), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:841-:d:1604470
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/5/841/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/5/841/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. A. A. Elmandouh & M. E. Elbrolosy & Ram Jiwari, 2022. "Integrability, Variational Principle, Bifurcation, and New Wave Solutions for the Ivancevic Option Pricing Model," Journal of Mathematics, Hindawi, vol. 2022, pages 1-15, September.
    2. Adel Elmandouh & Aqilah Aljuaidan & Mamdouh Elbrolosy, 2024. "The Integrability and Modification to an Auxiliary Function Method for Solving the Strain Wave Equation of a Flexible Rod with a Finite Deformation," Mathematics, MDPI, vol. 12(3), pages 1-15, January.
    3. Feiyun Pei & Guojiang Wu & Yong Guo, 2023. "Construction of Infinite Series Exact Solitary Wave Solution of the KPI Equation via an Auxiliary Equation Method," Mathematics, MDPI, vol. 11(6), pages 1-25, March.
    4. Abdulrahman Alomair & Abdulaziz S. Al Naim & Ahmet Bekir, 2024. "Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability," Mathematics, MDPI, vol. 13(1), pages 1-17, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yuli D. Chashechkin & Artem A. Ochirov, 2023. "Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field," Mathematics, MDPI, vol. 11(21), pages 1-18, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:841-:d:1604470. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.