Bifurcation Analysis, Chaotic Behavior, and Exact Traveling Wave Solutions for the (2 + 1)-Dimensional Complex Modified Korteweg–de Vries Equation

This paper examines the (2 + 1)-dimensional complex modified Korteweg–de Vries equation, which describes the intricate motion of water particles from the surface to the bottom. By applying an appropriate wave transformation that reduces the governing nonlinear partial differential equations to an ordinary differential system, we conduct a comprehensive bifurcation analysis that identifies equilibrium points and characterizes their phase-space properties. Under periodic perturbations, the system exhibits bifurcations, quasi-periodicity, multistability, and chaotic behavior, supported by Lyapunov exponents, time-series evolution, phase portraits, and Poincaré sections. To obtain exact soliton solutions of various types, we employ both the dynamical system method and the generalized double auxiliary equation method. The physical features of the solutions are depicted using 2D and 3D representations of their real, imaginary, and absolute components. A comparison with related studies shows that the proposed approaches not only reproduce previously reported solutions but also yield broader and more general soliton families.