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

Wensheng Chen, Jalil Manafian (), Khaled Hussein Mahmoud, Abdullah Saad Alsubaie, Abdullah Aldurayhim and Alabed Alkader
Additional contact information
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
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

Mathematics, 2023, vol. 11, issue 16, 1-35

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.

Keywords: Gilson–Pickering equation; standard tan( ? /2)-expansion technique; Paul–Painlevé approach; He’s variational direct technique; soliton solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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