Dynamical analysis, chaotic behavior and soliton solutions of the generalized unstable space–time fractional nonlinear Schrödinger equation

Abstract

The main objective of this work is to obtain soliton solutions, perform dynamical analysis for the generalized unstable space–time fractional nonlinear Schrödinger equation, and investigate chaotic dynamics of its perturbed reduced dynamical system. First, a traveling wave transformation is applied to convert the governing equation into a two‑dimensional planar dynamical system. Second, the bifurcation of the dynamical system is analyzed using the theory of planar dynamical systems. Third, by introducing a perturbation term to the dynamical system, chaotic behavior is demonstrated through two‑dimensional and three‑dimensional phase portraits. A sensitivity analysis of the dynamical system is then carried out using the Runge–Kutta method. Furthermore, chaos of the perturbed system is detected and characterized using recurrence plots, power spectra, Poincaré sections, fractal dimensions, time series of state variables, fractal phase‑space structures, and Lyapunov exponent curves. Finally, the planar dynamical system approach is employed to derive explicit dark and bright soliton solutions for the original fractional model.

Graphical abstract