$$\beta$$-Fractional $$(n+1)$$-dimensional generalized KP model: nonlinear dynamical behaviors, analytical wave structures, bifurcation, and sensitivity analysis

This study introduces an integrable generalization of the Kadomtsev–Petviashvili model in arbitrary spatial dimensions. The Kadomtsev–Petviashvili equation serves as a fundamental framework in describing a wide range of physical phenomena, including hydrodynamic wave disturbances, plasma dynamics, and nonlinear optical systems. In this work, the classical Kadomtsev–Petviashvili equation is modified to incorporate $$\beta$$-fractional derivatives, extending its applicability to more complex dynamical scenarios. By employing the extended tanh function method in conjunction with the Riccati differential equation, an analytical of exact solutions such as dark, singular, and periodic wave forms are derived. These solutions give valuable mathematical insight into wave propagation and offer significant physical relevance for practical applications in physics and engineering. Two-dimensional (2D) and three-dimensional (3D) plots of the obtained wave equations are drawn under certain values of the parameters and at different cases of $$\beta$$. In order to investigate further the system’s dynamical properties, the governing equations were transformed by means of the Galilean transformation. That way, a detailed investigation of the nonlinear structure was possible. A phase space analysis is carried out using planar dynamical system techniques in order to find bifurcations, chaotic behavior, and sensitivity of the system to initial conditions. An instability analysis due to the interaction of nonlinear and dispersive effects in the proposed model is carried out. The stability criterion for waves with regard to small perturbations was determined. Our results are contrary to the findings available in the literature, and this is the first attempt that has been made to study the Hamiltonian structure, bifurcation analysis, sensitivity analysis, and modulation instability for the model considered.