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Our aim is to examine the dynamic characteristics of the (3+1)-dimensional generalized equation governing shallow water waves. When the horizontal extent of the fluid significantly surpasses the vertical dimension, the employment of shallow water equations becomes appropriate. By employing an inventive Ricatti equation mapping approach, we have obtained a range of solitary wave solutions in both explicit and generalized forms. Solitons are particularly useful in signal and energy transmission due to their ability to preserve their shape during propagation. By studying soliton solutions in nonlinear systems, we can gain valuable knowledge applicable to practical fields. To deepen our grasp of the equation’s physical implications, we have presented some solutions through 3D, 2D, and contour graphics. Employing a linear stability approach, we confirm the equation’s stability. The method used distinguishes out for its simplicity, dependability, and ability to generate novel solutions for nonlinear partial differential equations in the area of mathematical physics. The research findings reported here demonstrate the viability of the used method in studying nonlinear phenomena in the studied equation as well as other nonlinear problems in mathematical physics. By employing this tool, academics have the chance to deepen and increase their grasp of the complex mathematical concepts underlying real-world problems.
Nasreen et al. (Wed,) studied this question.