Geometric Classification of Phase Orbits and Chaotic Dynamics in the Fractional Coupled (2+1)-Dimensional Nonlinear Schrodinger Equation

This paper investigates the dynamical behaviors and exact localized wave solutions of the conformable time-fractional coupled (2+1)-dimensional nonlinear Schrödinger equation. Studying this model helps in understanding complex nonlinear wave propagation; however, the phase-space structures and chaotic behaviors of such high-dimensional fractional systems require further exploration. To address this, a methodology combining geometric analysis and numerical simulation is employed. By applying a traveling-wave transformation, the fractional governing equation is reduced to a planar dynamical system. We utilize Hamiltonian phase-plane analysis and critical energy level evaluations to classify the phase orbits. Furthermore, direct time-series tracking, Poincaré sections, and the calculation of the maximum Lyapunov exponents are utilized to probe the dynamical response under external periodic perturbations. The main results include the derivation of several exact traveling wave solutions, such as smooth periodic waves, bell-shaped solitary waves, and singular traveling waves, which are derived corresponding to their distinct phase orbits. Additionally, numerical analyses demonstrate that external perturbations can break the system’s integrable structure and induce a transition into chaotic states. It is concluded that the topological structures of these localized waves are closely related to distinct Hamiltonian energy thresholds, and the disturbed system exhibits quantifiable sensitivity to initial conditions. Distinct from existing algebraic solving methods, the originality of this work lies in elucidating a physically interpretable connection between the topological classification of wave structures and the quantitative evaluation of chaotic dynamics.