Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method

Wafaa B. Rabie; Hamdy M. Ahmed; Taher A. Nofal; E. M. Mohamed; Wafaa B. Rabie; Hamdy M. Ahmed; Taher A. Nofal; E. M. Mohamed

doi:10.3934/math.20241593

AIMS Mathematics

2024, Volume 9, Issue 12: 33386-33400. doi: 10.3934/math.20241593

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Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method

1.
Department of Engineering Mathematics and Physics, Higher Institute of Engineering and Technology, Tanta, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
4.
Basic Science Department, Delta Higher Institute for Engineering and Technology, Egypt

Received: 03 September 2024 Revised: 27 October 2024 Accepted: 18 November 2024 Published: 25 November 2024
MSC : 35C07, 35C08, 35C09, 35G20

Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.
- Kac-Wakimoto equation,
- soliton solutions,
- hyperbolic solutions,
- wave profile
Citation: Wafaa B. Rabie, Hamdy M. Ahmed, Taher A. Nofal, E. M. Mohamed. Novel analytical superposed nonlinear wave structures for the eighth-order (3+1)-dimensional Kac-Wakimoto equation using improved modified extended tanh function method[J]. AIMS Mathematics, 2024, 9(12): 33386-33400. doi: 10.3934/math.20241593

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Abstract

Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.

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