Gross-Pitaevskii systems of fractional order with respect to multicomponent solitary wave dynamics.

This study presents a comparative analysis of the fractional Gross-Pitaevskii equation (GPE), a fundamental nonlinear Schrödinger-type model, focusing on the derivation of exact soliton solutions critical for understanding nonlinear phenomena such as superfluidity and superconductivity. We compare the effectiveness of β-fractional and M-truncated fractional derivatives in solving the complex fractional GPE. This comparison reveals that while both fractional derivatives enable the construction of diverse optical soliton solutions-including hyperbolic, periodic, Jacobi elliptic, and exponential forms-the β-derivative provides smoother soliton profiles with computational simplicity, whereas the M-truncated derivative captures richer oscillatory dynamics due to its enhanced memory effects. Employing advanced analytical tools-namely the generalized extended direct algebraic method (gEDAM) and the Kummar-Malik (KM) method-along with Wolfram Mathematica for verification, we extract and rigorously validate a variety of exact solutions. The generated solutions and their corresponding wave profiles under varying parameters highlight the distinct physical implications of each fractional derivative approach. Our results offer a robust framework for modeling nonlinear fractional dynamics, with applications spanning optical fibers, plasma physics, mathematical physics, and condensed matter systems. The comparative insights gained deepen the understanding of fractional-order effects in nonlinear wave evolution, paving the way for further exploration in complex physical systems.