Using the time-splitting spectral method, the time- dependent GPE is discretized further more and different forms of potentials are used to obtain interesting results for trapping in harmonic and different potentials, confined in this work. Spatial and time discretization error analyses for present results were compared with those obtained by other authors for different techniques. Fourth- order time-splitting pseudospectral (TSP), Hermite pseudospectral (HP), Laguerre pseudospectral (LP) and Laguerre-Hermite pseudospectral (LHP) methods are improved which are developed to study GPE of rotating BEC. In all cases the characteristic feature of vortex dynamics and interactions in superconductivity and superfluidity are studied by considering the Ginzburg-Landau-Schr”dinger equation (GLSE).Finally, to describe the dynamics of atom laser, the interaction Hamiltonian is derived between the two-level atoms with an electromagnetic field and then introducing the equations of motion. This interaction Hamiltonian is used to describe the outcoupling process in a Raman atom laser.