article · Arab Journal of Basic and Applied Sciences
This research investigates the generalized shallow water-like equation in three spatial dimensions plus time, which serves as a mathematical framework for studying wave dynamics in ocean physics. Using an expansion technique based on double variables, the study identifies several new propagating wave solutions. These newly derived mathematical expressions are formulated using hyperbolic, trigonometric, and rational functions. By varying the free parameters within these solutions, multiple soliton forms are established, including kink, anti-kink, anti-bell, singular, and single periodic structures. To support the interpretation of these complex dynamics, two- and three-dimensional graphical representations are generated under specific parameter conditions. These visual models clarify the general physical behaviour and geometry of the waves, contributing new theoretical solutions to the broader study of nonlinear wave phenomena.
Understanding how waves travel and interact in shallow water is essential for ocean physics. By identifying exact mathematical wave solutions and visualizing their shapes, this theoretical work helps researchers better describe and predict complex, nonlinear fluid movements that occur in marine environments.
The abstract describes purely theoretical mathematical research into wave equations and does not indicate an application pathway or commercialisation potential.
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The (3 1)-dimensional generalized shallow water equation is a significant mathematical framework for analyzing the dynamic behavior of waves in ocean physics. The purpose of this article is to investigate some more generic soliton solutions of the generalized shallow water-like model in three dimensions. The investigation is conducted utilizing the sophisticated mathematical methodology known as the double variables / 0 =/, 1=/ expansion technique. With this approach, we produce new propagating wave solutions for this model in the form of hyperbolic, trigonometric, and rational functions. In addition, we offer twoand three-dimensional graphical representations to help visualize the intricate physical phenomena of the system. We have constructed many soliton solutions, such as kink shape soliton solutions, anti-bell shape solutions, single periodic solutions, singular soliton solutions, and anti-kink shape solutions for different values of the free parameters involved in the obtained solutions. These graphical representations are predicated on certain parameter selections, which facilitate the understanding of the complicated general behavior for this model. Through the presentation of new findings in the field of soliton solutions for the aforementioned equation, this paper offers fresh perspectives and highlights hitherto overlooked aspects of this fascinating mathematical challenge. The paper illuminates new results on soliton solutions with different geometrical structures for the given equation, revealing hitherto overlooked facets of this intriguing mathematical challenge.
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DOI: 10.1080/25765299.2024.2313245
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