It is shown that the quasi-one-dimensional Bose-Einstein condensate is experimentally accessible and rich in intriguing phenomena. We demonstrate numerically and analytically the existence, stability and perturbation-induced dynamics of all types of stationary states of the quasi-one-dimensional nonlinear Schrödinger equation for both repulsive and attractive cases. Among our results are: the connection between stationary states and solitons; creation of vortices from such states; manipulation of such states with simple phase profiles; demonstration of the fragility of the condensate phase in response to shock; and a robust stabilization of the attractive Bose-Einstein condensate.
We explore how to control the dynamical instability of a quasi-one dimensional (1D) stationary state Bose-Einstein condensate (BEC) consisting of fixed
N atoms with arbitrary time-independent external potential and atomic scattering length. The dependence of the perturbed solutions on the velocity field, external potential, condensed atom number, and the initial perturbations is demonstrated analytically, that shows the stability, instability and undetermined stability of the system. Some sufficient conditions on the stability and controlling criteria to the instability are established. We take a BEC held in an optical lattice potential as an example establishing the stability regions on parameter space and illustrating the scheme for controlling instability. The results contain the known stability assertions for the lattice potential case and agree with the recent experimental findings qualitatively.