Nonlinear Dynamics of Bose-Condensed Gases by Means of a Low- to High-Density Variational Approach

We propose a versatile variational method to investigate the spatio-temporal dynamics of one-dimensional magnetically-trapped Bose-condensed gases. To this end we employ a q-Gaussian trial wave-function that interpolates between the low-and the high-density limit of the ground state of a Bose-condensed gas. Our main result consists of reducing the Gross-Pitaevskii equation, a nonlinear partial differential equation describing the T = 0 dynamics of the condensate, to a set of only three equations: two coupled nonlinear ordinary differential equations describing the phase and the curvature of the wave-function and a separate algebraic equation yielding the generalized width. Our equations recover those of the usual Gaussian variational approach (in the low-density regime), and the hydrodynamic equations that describe the high-density regime. Finally, we show a detailed comparison between the numerical results of our equations and those of the original Gross-Pitaevskii equation.