Nonlinear dynamics of Bose-condensed gases by means of a q-Gaussian 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 describes both the low- and the high-density limit of the ground state of a Bose-condensed gas. Unlike previous analytical models, we do not approximate the dynamics of the condensate as a dynamical rescaling of the initial density profile. Instead, we allow the shape of the condensate’s density profile to change in time. 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.