  The combination of multi-component Bose-Einstein condensates (BECs) and phase
imprinting techniques provides an ideal platform for exploring nonlinear
dynamics and investigating the quantum transport properties of superfluids. In
this paper, we study abundant density structures and corresponding dynamics of
phase-separated binary Bose-Einstein condensates with phase-imprinted single
vortex or vortex dipole. By adjusting the ratio between the interspecies and
intraspecies interactions, and the locations of the phase singularities, the
typical density profiles such as ball-shell structures, crescent-gibbous
structures, Matryoshka-like structures, sector-sector structures and
sandwich-type structures appear, and the phase diagrams are obtained. The
dynamics of these structures exhibit diverse properties, including the
penetration of vortex dipoles, emergence of half-vortex dipoles, co-rotation of
sectors, and oscillation between sectors. The pinning effects induced by a
potential defect are also discussed, which is useful for controlling and
manipulating individual quantum states.
