  We study analytically and asymptotically as well as numerically ground states
and dynamics of two-component spin-orbit-coupled Bose-Einstein condensates
(BECs) modeled by the coupled Gross-Pitaevskii equations (CGPEs). In fact, due
to the appearance of the spin-orbit (SO) coupling in the two-component BEC with
a Raman coupling, the ground state structures and dynamical properties become
very rich and complicated. For the ground states, we establish the existence
and non-existence results under different parameter regimes, and obtain their
limiting behaviors and/or structures with different combinations of the SO and
Raman coupling strengths. For the dynamics, we show that the motion of the
center-of-mass is either non-periodic or with different frequency to the
trapping frequency when the external trapping potential is taken as harmonic
and the initial data is chosen as a stationary state (e.g. ground state) with a
shift, which is completely different from the case of a two-component BEC
without the SO coupling, and obtain the semiclassical limit of the CGPEs in the
linear case via the Wigner transform method. Efficient and accurate numerical
methods are proposed for computing the ground states and dynamics, especially
for the case of box potentials. Numerical results are reported to demonstrate
the efficiency and accuracy of the numerical methods and show the rich
phenomenon in the SO-coupled BECs.
