Dynamical analysis and exact solitary wave solutions of a (3 + 1)-dimensional Davey–Stewartson system with parabolic law nonlinearity

This paper presents a comprehensive dynamical systems analysis of a generalized (3 + 1)-dimensional Davey–Stewartson equation with parabolic law nonlinearity. The complex partial differential equation is reduced to a planar dynamical system, enabling detailed bifurcation analysis, equilibrium point characterization, and complete phase portrait construction across different parameter regimes. External periodic forcing transforms the conservative system into a dissipative one, facilitating chaotic dynamics that are confirmed through Lyapunov exponent computation and sensitivity analysis. The complete Lyapunov spectrum provides quantitative evidence of chaotic behavior. Using the new mapping method and the new Kudryashov method, we obtain novel exact solutions, including bright and dark solitons, rogue waves, kink-shaped structures, and periodic patterns. The derivation of these exact solutions addresses the distinctive challenges posed by parabolic law nonlinearity and establishes crucial connections with phase space structures identified through bifurcation analysis. The interplay between analytical solutions and dynamical systems theory reveals fundamental mechanisms governing wave stability, bifurcations, and chaos in higher-dimensional nonlinear systems, with applications in nonlinear optics and fluid dynamics.