On the Ground States of Rotating Spin-Orbit Coupled Two-Component Bose-Einstein Condensates

In this paper, we study mathematically and numerically the ground states of rotating spin-orbit coupled two-component Bose-Einstein condensates modeled by the coupled Gross-Pitaevskii equations. Firstly, we rigorously prove the existence and nonexistence results for the ground states and derive some analytical properties, including the virial identity and negativity of SOC energy. Secondly, we propose an efficient and accurate preconditioned nonlinear conjugate gradient (PCG) algorithm to compute the ground states. We truncate the whole space into a bounded rectangular domain and readily apply the Fourier spectral method to approximate the wave function. The PCG method is successfully adapted with appropriate modifications to the adaptive step size control strategy for the one-parameter energy minimization problem and to the choice of preconditioners, achieving great performance in terms of accuracy and efficiency. Lastly, we carry out extensive numerical experiments to confirm the spatial spectral accuracy by traversing the most commonly-used initial guesses for each component. Moreover, we investigate the effects of local interaction, angular velocity, spin-orbit coupling and external trapping potential on the ground states, and predict some interesting physical phenomena, such as vortex lattices.