Novel Analytical Methods for and Qualitative Analysis of the Generalized Water Wave Equation

Haitham Qawaqneh, Abdulaziz S. Al Naim and Abdulrahman Alomair ()
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Haitham Qawaqneh: Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan, Amman 11733, Jordan
Abdulaziz S. Al Naim: Accounting Department, Business School, King Faisal University, Al-Ahsa 31982, Saudi Arabia
Abdulrahman Alomair: Accounting Department, Business School, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 14, 1-26

Abstract: For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved ( G ′ / G ) expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering.

Keywords: generalized water wave model; fractional derivative; analytical methods; exact solitons; stability analysis; modulation instability analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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