Enhanced analytical solutions of coupled conformable Gross-Pitaevskii equations incorporating external potentials using a residual series method

This work concerns the construction of approximate analytical solutions for the nonlinear complex conformable Gross-Pitaevskii equations with an external potential using the residual series method in conformable sense. This technique combines the flexibility of residual error function and generalized multivariable power series, utilizing time-dependent conformable derivatives. By minimizing the residual error, solitary wave solutions in a subtle pattern are generated. Further, convergence analysis is provided to illustrate the theoretical framework of our scheme in handling the proposed nonlinear models. Therefore, for practical computation, several naturalistic applications for Bose-Einstein condensates involving zero trapping, periodic boxes, optical lattices, and harmonic potentials are examined. Additionally, numerical computations and graphical representations are provided to verify the correctness and accuracy of the tested applications. Moreover, the dynamic behaviors of wave soliton solutions are captured at different parameters. These solutions demonstrate the effectiveness of the method and its ease of use in solving many complex nonlinear partial differential equations arising in quantum optics, quantum gases, quantum fluids, and other states of quantum mechanics.