A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation

Yousef Jawarneh; Humaira Yasmin; Ali M. Mahnashi; Yousef Jawarneh; Humaira Yasmin; Ali M. Mahnashi

doi:10.3934/math.20241678

AIMS Mathematics

2024, Volume 9, Issue 12: 35308-35325. doi: 10.3934/math.20241678

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A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

Received: 20 September 2024 Revised: 04 December 2024 Accepted: 06 December 2024 Published: 18 December 2024
MSC : 34G20, 35A20, 35A22, 35R11

In this paper, we use the Riccati–Bernoulli sub-ODE method in conjunction with the Bäcklund transformation to find out the exact solutions of the nonlinear time–space fractional Bogoyavlenskii equation. The obtained solutions encompass multiple kink solitary wave solutions that are quite unique and important in addition to solutions presented in hyperbolic, trigonometric, and rational function forms. This equation describes central factors influencing its behavior including fluid dynamics in shallow water waves and plasma, which demonstrates our conclusions have broad applications for such systems. We also study the effect of the fractional order parameter ($ \alpha $) on solutions and plot their behavior using MATLAB in two dimensions. This work also contributes to the knowledge of the physical structures of the fractional Bogoyavlenskyi equation apart from showcasing the potential of the Riccati–Bernoulli sub-ODE method when applied to nonlinear fractional differential equations.
- fractional Bogoyavlenskii equation,
- Bäcklund transformation,
- non-linear differential equations,
- exact solutions
Citation: Yousef Jawarneh, Humaira Yasmin, Ali M. Mahnashi. A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation[J]. AIMS Mathematics, 2024, 9(12): 35308-35325. doi: 10.3934/math.20241678

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Abstract

In this paper, we use the Riccati–Bernoulli sub-ODE method in conjunction with the Bäcklund transformation to find out the exact solutions of the nonlinear time–space fractional Bogoyavlenskii equation. The obtained solutions encompass multiple kink solitary wave solutions that are quite unique and important in addition to solutions presented in hyperbolic, trigonometric, and rational function forms. This equation describes central factors influencing its behavior including fluid dynamics in shallow water waves and plasma, which demonstrates our conclusions have broad applications for such systems. We also study the effect of the fractional order parameter ($ \alpha $) on solutions and plot their behavior using MATLAB in two dimensions. This work also contributes to the knowledge of the physical structures of the fractional Bogoyavlenskyi equation apart from showcasing the potential of the Riccati–Bernoulli sub-ODE method when applied to nonlinear fractional differential equations.

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