Nonlinear wave structures and sensitivity analysis in the higher-order generalized KP–Boussinesq equation

In this work, we present a comprehensive analytical investigation of the (3+1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation, a higher-dimensional nonlinear evolution equation of significant relevance to fluid dynamics and wave propagation. By employing two robust analytical techniques, the enhanced modified extended tanh expansion method and the unified method, we construct a diverse class of exact solutions, including bright, dark, periodic, and rational wave structures. These solutions are illustrated using three-dimensional surface plots, two-dimensional profiles, and contour maps to capture their dynamic behaviors and physical characteristics. Furthermore, a detailed bifurcation analysis is conducted to investigate the system’s qualitative dynamics under varying parameter regimes. Phase portraits are employed to illustrate transitions between equilibrium states and to characterize the stability and bifurcation structures. A sensitivity analysis is also performed to assess the model’s robustness with respect to parameter variations. Overall, the findings contribute new insights into the nonlinear dynamics and integrability properties of higher-dimensional evolution equations, broadening the current understanding of complex wave phenomena.