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The use of Rydberg dressing technology to achieve long-range soft-core interaction in Bose Einstein condensates opens up new avenues for exploration of supersolid and its related phenomenon. We investigate the ground state of a two-component spin-orbit-coupled Bose-Einstein condensates with both long-range soft-core and contact interactions in radially periodic potentials. Our results show that the ground-state structures of the system are strongly influenced by spin-orbit coupling, contact interactions, long-range soft-core interactions, and the amplitude of the external potential. We find that such parameters can been used to induce desired ground-states structures, such as necklace structure of lump, striped standing wave, and especially ring dark soliton. Furthermore, we observe
that the long-range soft-core interactions are used to manipulate the transition between miscible immiscible phases like contact interactions. Our research provides another degree of freedom for manipulating supersolids in spin-orbit-coupled Bose-Einstein condensates.

The dynamical behaviors of a quasi-2D Bose-Einstein condensate (BEC) with three-body interactions through a moving obstacle potential are studied numerically. Various vortex structures are discovered under different strength of three-body interactions when the two-body interaction is attractive or repulsive. When the width and moving velocity of the obstacle potential reach critical value, periodic anti-symmetric double-row vortex pairs will be released alternately in BEC, and a B´en´ard-von K´arm´an (BvK) vortex street will be formed eventually. We noticed that the BvK vortex street can be excited when the three-body interactions are taken into account even if the two-body interaction is attractive. The mean value of the distance between two vortex rows to the distance between two vortex pairs in the same row is about 0.2. It is slightly smaller than the stability condition 0.28 without considering the three-body interaction. The parameter regions of vortex patterns at different three-body interaction are determined. It is found that an appropriate value of three-body interaction with larger velocity and lesser width is favorable to the formation of BvK vortex street. In a pair of point vortices, the distance and angular velocity between them are nearly invariant while they rotate around their center. The internal rule of vortex pair are also analyzed by calculating the drag force acting on the obstacle potential. Finally, we proposed an experimental protocol to realize the vortex street in 87Rb BEC with three-body interactions.