Conservation Laws and Dynamics of Solitons to the Multicomponent Fractional Gross–Pitaevskii System: Application in the Bose–Einstein Condensation

In this work, we study the β-fractional multicomponent Gross–Pitaevskii (G–P) system, which plays an important role in modeling Bose–Einstein condensates (BECs) and propagation in nonlinear waves. In Bose–Einstein condensation, the G–P equation is of great importance as it describes the condensate wave function’s behavior. These models are very significant due to their extensive range of applications in numerous scientific and technical domains, particularly nonlinear optical fibers, processing of signal via fibers optic, and magnetic field. The generalized Arnous method (GAM) and multivariate generalized exponential rational integral function (mGERIF) technique are under consideration to investigate the dynamics of different wave profiles to the multicomponent G–P system emerging in the Bose–Einstein condensation mechanism. To attain different types of exact results, the main model is transformed into a nonlinear ordinary differential equation by employing a complex wave transformation with β-fractional derivatives. The attained results are novel and constitute a variety of solutions which have not been reported before for the considered governing model. Furthermore, we also display 3D, 2D, and contour structures giving the forms of novel pursued solitary waves by assigning different numerics to the parameters for making best understand of the complex phenomena. We also derive the conservation laws of mass, magnetization, and energy for the dynamical system, confirming its Hamiltonian structure and ensuring the physical consistency of the model. In contrast with the existing methodologies, it represents that the suggested approaches are powerful and efficient. The calculating exertion and achieved outcomes shows that proposed techniques are helpful to resolve various other nonlinear problems arising in applied sciences and the obtained solutions play a significant role in the investigation of wave propagation and related disciplines. All extracted solutions are evaluated by back substitution to the actual examined problem with the aid of software package Mathematica.