Dynamical behavior of analytical solutions and bifurcation analysis for a novel structured (2+1)-dimensional...

This work represents a novel integration of both frameworks in a single analytical context by using the modified extended (ME) mapping approach to get new, exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili and Shallow Water Wave equations. This model is essential for simulating multidimensional wave propagation processes, and it has several applications in shallow water hydrodynamics, plasma physics, nonlinear optics, and the investigation of long internal waves in stratified fluids. Bright soliton, dark soliton, singular soliton, and singular periodic, periodic, hyperbolic, exponential, Jacobi elliptic (JE), and Weierstrass elliptic doubly periodic solutions are among the many unique structures that are constructed in this study. The dynamical resilience of the generated solutions is evaluated by means of a linear stability study and bifurcation analysis. Crucially, the findings provide fresh information on nonlinear wave collision processes by demonstrating that the soliton interactions either behave elastically or preserve the soliton under particular conditions. The spatiotemporal dynamics and interactions of the resulting solutions are shown through a succession of 2D, 3D, and contour plots, which enhance the findings' physical understanding and application. A new theoretical understanding of nonlinear wave propagation in fluids and related physical systems is provided by this work.