Exploring Jacobi Elliptic and Periodic Solitary Wave Solutions for the Family of 3-D WBBM Equations Through the Generalized Approach

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Abstract
In this paper, we analyze the family of three-dimensional Wazwaz-Benjamin-Bona-Mahony (3-D WBBM) equations using the generalized Jacobi elliptic function expansion method. Understanding complicated wave patterns is critical, therefore, the family of 3-D WBBM equations is a valuable mathematical model with application in fluid mechanics, plasma dynamics, biomechanics, and engineering sciences. The main objective of this study is to build several new multiple wave form solutions for the nonlinear three-dimensional Wazwaz-Benjamin-Bona-Mahony (3-D WBBM) equations. By first using the travelling wave transformation, the nonlinear partial differential equation is transformed into an ordinary differential equation. Then the generalized Jacobi elliptic function expansion method is used to execute a finite series expansion of degree n. Due to this analytical method, we obtained numerous new exact solutions, including periodic solutions in Jacobi elliptic function forms and their corresponding Solitary and shock wave solutions in a limit convergence sense when the modulus parameter m approaches 1 and 0. These results are particularly useful for nonlinear science and mathematical physics professionals, since they provide important insights into the structure and evolution of nonlinear waves in a variety of physical circumstances. To improve the physical description of the solutions, several typical wave profiles are offered to provide a comprehensive analysis of the wave characteristics of the solutions in 2-D, 3-D, and contour visualizations were generated using accurate parameters value with the help of Mathematica. Moreover, by generating novel and accurate propagating soliton waveform solutions, the generalized Jacobi elliptic function expansion methods highlight its importance in uncovering key aspects of the model behaviours as well as suggesting potential applications in the study of water waves.

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