We study ground states of a Bose-Einstein condensate in a harmonic plus Gaussian trap. It is found that as the condensate forms a giant vortex the number of vortices is equal to the average angular momentum and the density distribution of the ground state is the same as that of angular momentum. We draw a conclusion that the ground state with giant vortex is the eigen state of angular momentum. As the potential well changes from an isotropy toroidal trap to an anisotropic toroidal trap,the ratio of the average angular momentum of the condensates to the number of vortices decreases slowly from 1, and then drops rapidly and stays near 0.5. Characteristics of density distribution of condensate and angular momentum distribution are given and explanation is shown.

Thomas-Fermi approximation (TFA) and imaginary-time propagation method are used to study ground states of rotating Bose-Einstein condensates in an annular trap. Ground state density profiles of condensates experience a transition from vortex lattice phase to giant vortex phase with increase of angular frequency or with increase of width and center height of trap potential. Particularly, ground state density profiles change from a disc shape into an annulus shape with the increase of width and center height of trap potential, when angular frequency is zero. Finally, comparison between ground state density profiles, obtained by analytical method and numerical method is made. They coincide with each other.

Generation and properties of dark solitons are studied in detail by means of numerical simulation of quasi one-dimensional heteronuclear two-component Bose-Einstein condensates in syntonic potential. Dark solitons are only induced in one of the condensate components by modulational instability with instantaneous conversion of repulsive interspecies interaction to attraction. Solitons pass through each other periodically in syntonie potential. In addition, the number of dark solitons is affiliated with ratio of particle mass and ratio of particles numbers.