Nonlinear dynamics and bifurcation analysis of integrable models for long wave propagation in plasma physics and oceanography

Abstract In this paper, the newly developed Kudryashov method is used for obtaining wave solutions of considered equations which are the (3+1) dimensional Kadomtsev–Petviashvili equation (KP) and the generalized Broer–Kaup–Kupershmidt system. For this purpose, the equations are reduced to ordinary differential equations via wave transformations. Then, according to the homogeneous balance method, the balance numbers are calculated. After these calculations, the auxiliary solutions are proposed and substituted into the ordinary differential equations. Systems of determining equations are obtained and solved for the arbitrary constants. Finally, solutions are derived and classified as bright and singular solitons. Three dimensional, two dimensional and contour plots are generated for some solutions via Maple software. These plots are an important tools in determining the physical meaning of the solutions. Furthermore, a comprehensive analysis of bifurcation, sensitivity, periodic and quasi-periodic behaviors of the KP equation is conducted. To the best of the authors’ knowledge, the applied methods are used for the first time on these equations, and the solutions obtained are novel. These findings offer significant contributions to applied mathematics, physics, and related scientific fields.