Qualitative Analysis for Modifying an Unstable Time-Fractional Nonlinear Schrödinger Equation: Bifurcation, Quasi-Periodic, Chaotic Behavior, and Exact Solutions

This work explores the qualitative dynamics of the modified unstable time-fractional nonlinear Schrödinger equation (mUNLSE), a model applicable to nonlinear wave propagation in plasma and optical fiber media. By transforming the governing equation into a planar conservative Hamiltonian system, a detailed bifurcation study is carried out, and the associated equilibrium points are classified using Lagrange’s theorem and phase-plane analysis. A family of exact wave solutions is then constructed in terms of both trigonometric and Jacobi elliptic functions, with solitary, kink/anti-kink, periodic, and super-periodic profiles emerging under suitable parameter regimes and linked directly to the type of the phase plane orbits. The validity of the solutions is discussed through the degeneracy property which is equivalent to the transmission between the phase orbits. The influence of the fractional derivative order on amplitude, localization, and dispersion is illustrated through graphical simulations, exploring the memory impacts in the wave evolution. In addition, an externally periodic force is allowed to act on the mUNLSE model, which is reduced to a perturbed non-autonomous dynamical system. The response to periodic driving is examined, showing transitions from periodic motion to quasi-periodic and chaotic regimes, which are further confirmed by Lyapunov exponent calculations. These findings deepen the theoretical understanding of fractional Schrödinger-type models and offer new insight into complex nonlinear wave phenomena in plasma physics and optical fiber systems.