Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

Abdulla-Al- Mamun; Chunhui Lu; Samsun Nahar Ananna; Md Mohi Uddin

doi:10.1038/s41598-024-55215-1

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

Sci Rep. 2024 Mar 18;14(1):6455. doi: 10.1038/s41598-024-55215-1.

Authors

Abdulla-Al- Mamun^{1

2

3}, Chunhui Lu^{4

5}, Samsun Nahar Ananna⁶, Md Mohi Uddin⁷

Affiliations

¹ College of Hydrology and Water Resources, Hohai University, Nanjing, 210098, People's Republic of China. abdullamamun21@gmail.com.
² State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, People's Republic of China. abdullamamun21@gmail.com.
³ Department of Computer Science and Engineering, Northern University of Business and Technology Khulna, Khulna, 9100, Bangladesh. abdullamamun21@gmail.com.
⁴ College of Hydrology and Water Resources, Hohai University, Nanjing, 210098, People's Republic of China.
⁵ State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, People's Republic of China.
⁶ School of Mathematics, Hohai University, Nanjing, 210098, People's Republic of China.
⁷ College of Water Conservancy and Hydropower, Hohai University, Nanjing, 210098, People's Republic of China.

Abstract

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Keywords: Exact solution; Lump shape; Sine-Gordon equation; Soliton shape; The rational sine-Gordon expansion method; Wazwaz-Benjamin-Bona-Mahony equation.