Novel Soliton and Periodic Wave Solutions of the (3+1)-Dimensional Shallow Water Wave Equation with Bifurcation Analysis

Abstract

Abstract This study derives novel exact traveling wave solutions for the $$(3 + 1)-$$ dimensional shallow water wave equation–a pivotal model in coastal hydrodynamics for tsunami prediction and tidal analysis. By employing an enhanced tanh-function method, we obtain a diverse spectrum of solutions, including dark, singular, and periodic solitons, as well as hyperbolic, Jacobi elliptic, rational, and exponential forms, which surpass the variety and generality reported in previous studies. These solutions uncover previously unexplored wave propagation patterns and interaction dynamics. A comprehensive bifurcation analysis elucidates the stability and phase transitions of the wave solutions, providing deeper analytical insight into their behavior. High-resolution graphical visualizations quantitatively demonstrate wave amplification and nonlinear interactions, confirming the superiority of our method in capturing complex physical phenomena. The results not only advance nonlinear wave theory but also enhance predictive models for marine hazard prevention and environmental monitoring strategies.