Dynamics of Attractive Bose-Einstein Condensates Trapped in Harmonic Potentials

Dynamics of attractive Bose-Einstein condensates trapped in harmonic potentials is studied. By use of the variance identity and a rigorous inequality, we obtain a sufficient condition for the collapse of the wavefunction in the nonlinear Schr?dinger equation (Gross-Pitaevskii equation). Further, by applying this method, we investigate the dynamic stability of a trapped Bose-Einstein condensate and its dependence on the initial shape of the condensate. The result agrees qualitatively well with that of the time-dependent variational method. We find that the anisotropy introduced in the initial state has a significant effect on the stability.