[[abstract]]This thesis concerns the ground-state property and dynamics of one- and two- component trapped Bose-Einstein condensates (BEC) in a variety of states and regimes. Variational method and finite-difference numerical method are applied throughout this thesis. Starting from the conditions of energetic sta-bility, coupled time independent and dependent Gross-Pitaveskii (GP) equa-tions are re-derived for a two-component system. With the phenomenon of phase separation built in, we introduce a trial wavefunction, called "modified Gaussian (MG) function". MG function is shown to be more suitable for a two-component as well as one-component system, providing that the (nonlin-ear) interaction effect is not too strong. Using MG trial wavefunction, the equilibrium and dynamical properties of a two-component system are studied in details. With the MG trial wavefunction in hand, we then study a BEC system of strong dipolar interaction. Since dipolar interaction is long-range and can be tuned to be resonant, a more realistic treatment for scattering should go beyond the first Born approximation (FBA). It is shown that the effect going beyond FBA is significantly enhanced when the system is close the phase boundary of collapse. To simulate the environment of a real crystalline solid, we also consider a one-dimensional optical lattice with a basis, i.e., a superlattice. Analytical results of acoustic and optical phonons are reported. Measurements of these modes can give unambiguous evidence to see whether the system is in the superfluid or Mott insulting regimes. Finally, we consider the effect of anharmonic trap on vortex arrays of a one-dimensional rapid ro-tating BEC. It is shown that due to the anharmonic quartic trap, the system remains stable at high rotating velocity and vortex lattices form even in the absence of the repulsive s-wave interaction (g). When g is present, the in-terplay between g and the quartic trap potential can lead to rich vortex lattice transition states as a function of ?, to which vortex lattices vanish eventually at some higher ?.
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