Analytic solutions for a (3+1)-dimensional extended shallow water wave equation in nonlinear phenomena

The study of analytic solutions for high-dimensional nonlinear wave equations is crucial for comprehending complex nonlinear phenomena in fluid dynamics. In this work, several novel analytic solutions and their dynamical behaviors are investigated by the extended homoclinic test approach and the improved $(G^{\prime }/G)$-expansion method for the (3+1)-dimensional extended shallow water wave equation. Based on the Hirota bilinear formulation, various breather wave solutions are constructed via the extended homoclinic test approach. Furthermore, multiple analytic solutions consisting of rational functions, trigonometric functions, and hyperbolic functions are derived using the improved $(G^{\prime }/G)$-expansion method. The features of the obtained solutions are illustrated through numerical simulations, including the shape-preserving properties during propagation of the soliton solutions and the divergence characteristics of rogue wave solutions. Furthermore, the relationships between soliton stability and rogue wave generation conditions are given through parameter variation analysis.