We consider the spin-1 Bose-Einstein condensates with the isotropic Rashba spin-orbit coupling in a two-dimensional torodial trap. Three types of striped phases are found in a non-rotational system, i.e., the stripe phase with the periodic density modulation along the azimuthal direction, the stripe phase with the periodic density modulation along both the azimuthal and the radial directions and the stripe phase with the periodic density modulation along the radial direction. By adding the rotation, the condensates occupy the mF = 0 component for small rotational frequency while occupy both the mF = 1 and mF = −1 components for large rotational frequency when both the relative interaction and the spin-orbit coupling are weak. For the stronger relative interaction and spin-orbit coupling, the vortices of the system are elongated along the radial direction and linked one after another. As the rotational frequency further increases, the density evolves from the elongated effect of the vortices into a laminar vortex ring.

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We study the ground-state phases of a two-dimensional rotating spin-orbit–coupled Bose-Einstein condensate loaded in a one-dimensional spin-dependent optical lattice. In the absence of rotation, the system undergoes a phase transition from the stripe phase to the supersolid phase gradually for weak spin-orbit coupling strength and directly for strong spin-orbit coupling strength as the depth of the optical lattice increases. With increasing the rotation strength, the stripe phase transforms to the ringlike state, and the domain number of the ringlike state increases when the Bose-Einstein condensate is in the weak depth of the optical lattice. The supersolid phase transforms to the phase with a single domain of a vortex line for small rotation strength, and alternating vortex lines along the x-direction appear for large rotation strength when the Bose-Einstein condensate is in the strong depth of the optical lattice. The spin textures of the ground-state phases are also discussed.

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