Dynamical behaviour and solutions in the fractional Gross–Pitaevskii model

The Gross–Pitaevskii equation is widely known for its applications in fields such as Bose–Einstein condensates and optical fibres. This study investigates the dynamical behaviour of various wave solutions to the M-fractional nonlinear Gross–Pitaevskii equation. Soliton solutions are derived using the GREFM, and the results are illustrated through 3D surface and contour plots. Furthermore, we apply the Galilean transformation to convert the second-order ordinary differential equation into a dynamical system and conduct bifurcation and sensitivity analyses. A perturbation term is introduced into the system to investigate the chaotic behaviour, Poincaré and time series analysis. The presence of chaos is further confirmed by using fractal dimension, return maps, power spectra and Lyapunov exponents. Additionally, we investigate multistability by analyzing the system’s response to various initial conditions. These contributions support the development of robust simulation tools for analyzing fractional dynamical systems in nonlinear optics and quantum mechanics.