Bifurcation and Exact Traveling Wave Solutions for the (3+1)-Dimensional Gardner-KP Equation

This paper investigates the bifurcation structures and exact traveling wave solutions of the (3+1)-dimensional Gardner–Kadomtsev–Petviashvili (Gardner–KP) equation, which generalizes the classical KdV, mKdV, and KP models by incorporating both quadratic and cubic nonlinearities along with weak transverse effects. Through an appropriate traveling wave transformation, the Gardner–KP equation is reduced to a two-dimensional planar dynamical system possessing a Hamiltonian structure. The bifurcation behavior of the equilibrium points and the corresponding phase portraits are systematically analyzed in the parametric plane. Using the bifurcation theory of dynamical systems, various types of orbits–homoclinic, heteroclinic, and periodic–are identified, corresponding respectively to solitary wave, kink (anti-kink), and periodic wave solutions. Exact analytical expressions of these traveling wave solutions are then derived in parametric forms for both the focusing and defocusing cases. The results demonstrate the rich dynamical behavior and diverse nonlinear wave phenomena exhibited by the Gardner–KP model, offering a comprehensive framework for understanding multidimensional nonlinear dispersive waves.