Stability analysis and continuation for the coupled Gross–Pitaevskii equations

We study linear stability analysis for the coupled Gross–Pitaevskii equations (CGPEs), which are used as the governing equations for rotating two-component Bose–Einstein condensates (BEC), and (ultrarapidly) rotating two-component Rydberg-dressed BEC. The CGPEs describe the coexistence of superfluidity and crystallization of BEC simultaneously. We show numerically that the linear stability of discrete steady state solutions is related to the angular velocity. Next, we describe some continuation algorithms for computing the ground state solutions of the CGPEs. Specifically, one of them can efficiently handle the case when long-range intra-component interactions of the two components are different. For ultrarapid rotation our numerical experiments demonstrate a competition between the supersolid crystal structure and the rotating-induced vortex lattice which gives rise to new phases.