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Cutting-Edge Analytical and Numerical Approaches to the Gilson–Pickering Equation with Plenty of Soliton Solutions

Author

Listed:
  • Wensheng Chen

    (Normal College, Ji Mei University, Xiamen 361021, China)

  • Jalil Manafian

    (Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz 5166616471, Iran
    Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov Str., Lankaran AZ4200, Azerbaijan)

  • Khaled Hussein Mahmoud

    (Department of Physics, College of Khurma University College, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia)

  • Abdullah Saad Alsubaie

    (Department of Physics, College of Khurma University College, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia)

  • Abdullah Aldurayhim

    (Mathematics Department, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia)

  • Alabed Alkader

    (Department of Sustainable Development Finance, Plekhanov Russian University of Economics, Moscow 117997, Russia)

Abstract

In this paper, the Gilson–Pickering (GP) equation with applications for wave propagation in plasma physics and crystal lattice theory is studied. The model with wave propagation in plasma physics and crystal lattice theory is explained. A collection of evolution equations from this model, containing the Fornberg–Whitham, Rosenau–Hyman, and Fuchssteiner–Fokas–Camassa–Holm equations is developed. The descriptions of new waves, crystal lattice theory, and plasma physics by applying the standard tan ( ϕ / 2 ) -expansion technique are investigated. Many alternative responses employing various formulae are achieved; each of these solutions is represented by a distinct plot. Some novel solitary wave solutions of the nonlinear GP equation are constructed utilizing the Paul–Painlevé approach. In addition, several solutions including soliton, bright soliton, and periodic wave solutions are reached using He’s variational direct technique (VDT). The superiority of the new mathematical theory over the old one is demonstrated through theorems, and an example of how to design and numerically calibrate a nonlinear model using closed-form solutions is given. In addition, the influence of changes in some important design parameters is analyzed. Our computational solutions exhibit exceptional accuracy and stability, displaying negligible errors. Furthermore, our findings unveil several unprecedented solitary wave solutions of the GP model, underscoring the significance and novelty of our study. Our research establishes a promising foundation for future investigations on incompressible fluids, facilitating the development of more efficient and accurate models for predicting fluid behavior.

Suggested Citation

  • Wensheng Chen & Jalil Manafian & Khaled Hussein Mahmoud & Abdullah Saad Alsubaie & Abdullah Aldurayhim & Alabed Alkader, 2023. "Cutting-Edge Analytical and Numerical Approaches to the Gilson–Pickering Equation with Plenty of Soliton Solutions," Mathematics, MDPI, vol. 11(16), pages 1-35, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3454-:d:1213957
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    References listed on IDEAS

    as
    1. Kai, Yue & Li, Yaxi & Huang, Liuke, 2022. "Topological properties and wave structures of Gilson–Pickering equation," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
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    3. Xiaoliang Xie & Linglu Huang & Stephen M. Marson & Guo Wei, 2023. "Emergency response process for sudden rainstorm and flooding: scenario deduction and Bayesian network analysis using evidence theory and knowledge meta-theory," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 117(3), pages 3307-3329, July.
    4. Turgut Ak & Asit Saha & Sharanjeet Dhawan, 2019. "Performance of a hybrid computational scheme on traveling waves and its dynamic transition for Gilson–Pickering equation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-17, April.
    5. Baoyong Guo & Huanhe Dong & Yong Fang, 2019. "Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation," Complexity, Hindawi, vol. 2019, pages 1-9, October.
    6. Turgut Ak & Asit Saha & Sharanjeet Dhawan, 2019. "Performance of a hybrid computational scheme on traveling waves and its dynamic transition for Gilson–Pickering equation," Surface Review and Letters (SRL), World Scientific Publishing Co. Pte. Ltd., vol. 30(04), pages 1-17, April.
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