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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 existence and nonexistence 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 nonperiodic or with different frequency than the trapping frequency when the external trapping potential is taken as harmonic and/or 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...

Ground States and Dynamics of Spin-Orbit-Coupled Bose--Einstein Condensates