Abundant Explicit Non-Traveling Wave Solutions to the \((3 + 1)\)-Dimensional Nonlinear Evolution Equation Using a Generalized Variable Separation Method

Non-traveling wave solutions allow us to characterize more complex natural phenomena across various scientific fields, which may not be easily analyzed using soliton solutions. This study introduces a novel approach for deriving a wide variety of explicit non-traveling wave solutions to the (3 + 1)-dimensional nonlinear evolution equation, utilizing a modified generalized variable separation technique. The newly obtained solutions encompass different non-traveling forms, including periodic solitary waves and soliton-like structures. These results underscore the efficacy of the method in tackling complex nonlinear partial differential equations. The findings presented in this article constitute innovative contributions to the equation modeling the velocity of water waves on the surface of shallow water. Furthermore, our employed technique can also provide a foundation for investigating the stability, dynamics, and practical applications of solutions to other similar problems.