This study develops a detailed Lie symmetry analysis for the one-dimensional time-fractional Gross–Pitaevskii equation (TFGPE), emphasizing how the adopted fractional derivative-Riemann-Liouville (RL) versus Caputo-modifies the symmetry algebra, nonlocal conservation laws, and invariant solution families. The complex-valued model is rewritten as an equivalent coupled system of real partial differential equations, and the corresponding infinitesimal generators are derived in a unified and systematic manner for both fractional formulations. We show that, in the Caputo setting, the standard global ????(1) phase invariance is retained, whereas in the RL formulation this key symmetry is destroyed because the associated initial data are not invariant and the fractional operator does not transform covariantly. Leveraging the resulting symmetry groups, we obtain similarity reductions and construct group-invariant solutions, and we further establish nonlocal conservation laws by applying Ibragimov’s nonlinear self-adjointness approach. Overall, our findings indicate that the selection of the fractional derivative is not a purely technical choice; it decisively affects physical consistency, the admissible symmetry structure, and the conservation behavior of fractional quantum models. The analysis offers a direct RL-Caputo comparison and supports the Caputo derivative as the more suitable framework for preserving the inherent symmetries of quantum systems.
Parastoo Kabi-Nejad Kabi-Nejad (Tue,) studied this question.