Analytical solutions and chaotic dynamics of the extended KP–Boussinesq model via phase diagnostics

This study investigates the dynamical behavior of a three-dimensional extended Kadomtsev-Petviashvili-Boussinesq (eKP-BO) equation, a higher-dimensional wave system that unifies Kadomtsev-Petviashvili-type weak dispersion with Boussinesq-type bidirectional propagation. Using the Formula: see text-expansion method, we systematically construct diverse classes of exact solutions of hyperbolic, trigonometric, and rational types. These analytical results enrich the family of admissible solitary and lump-type wave structures and provide closed-form benchmarks for validating numerical simulations. Uncover the underlying dynamics by reducing the governing equation to a planar system. This can then be analyzed through phase portraits, bifurcation diagrams, Poincaré sections, and time series. The results will reveal equilibrium structures such as centers, saddles, and cusps, as well as transitions from regular oscillations to period-doubling and fully developed chaos under period forcing. The study will also highlight both the advantages and limitations of the exponential expansion method, focusing on its ability to generate rich explicit solutions while noting its dependence on balance conditions. In sum, the results shed light on the nonlinear dispersive wave dynamics of the eKP-BO model in the unified framework and identify several potential future research paths, such as fractional-order models, stochastic perturbations, and hybrid analytical-numerical methods.