Analytical soliton solutions for the beta fractional derivative Gross-Pitaevskii system with linear magnetic and time dependent laser interactions

The local conformable beta Atangana derivative will be considered for the introduction of the fractional Gross-Pitaevskii model with conformable derivatives of beta type. Analytical expressions for soliton solutions are constructed by sub-equation method with elliptical functions. The main goal of the current research is to determine the general behavior of the soliton solutions, their dependence on the elliptical parameter and the influence of the fractional order parameter on the time and space scales of the solutions. New entire family of solitons were determined by considering the arising constrains over the parameters of the nonlinear fractional Gross-Pitaevskii system. The analytical expressions for the soliton solutions constructed for the fractional order case reduce to the well known solitons previously reported for hyperbolic and periodic tan-type singular solutions for the integer order limit value, when special cases of the Jacobi elliptic functions are considered. Solitons properties are depicted in 3-D level and 2-D illustrations. The fractional solitons here introduced possess some interesting time evolution behavior observed in the 3-D representations, these time properties are not present in the integer order case and has an important dependency on the fractional parameter of the beta derivative. The solitons here introduced for the nonlinear fractional Gross-Pitaevskii equation will be very useful in future works where additional interactions will be introduced for the study of different Bose-Einstein condensation phenomena, the coupled quasi-one dimensional Gross- Pitaevskii equation or other nonlinear phenomena where non regular oscillations will be involved.