An Efficient Time-Splitting Method to Simulate the Dynamics of Spin-Orbit Coupled spin-1 Bose-Einstein Condensates

In this paper, an efficient time-splitting Fourier spectral method is proposed to simulate the dynamics of spin-orbit coupled spin-1 Bose-Einstein condensates (SOC spin-1 BECs). We split the Hamiltonian into a linear part, which consists of the Laplace and SOC terms, and a nonlinear part that includes all the remaining terms. The linear subproblem is integrated analytically in phase space by solving an ordinary differential system of constant coefficient matrix. While, for the nonlinear subproblem, it is proved the coefficient matrix is actually time-independent in physical space, therefore, the nonlinear subproblem can be integrated exactly. Based on such two-step splitting, we construct high-order schemes to simulate the dynamics. Our method is spectrally accurate in space and high order in time. It is efficient, explicit, unconditionally stable and simple to implement. In addition, we derive some dynamical properties for SOC spin-1 BECs. Extensive numerical results are presented to confirm the accuracy and efficiency, illustrate the dynamical properties at discrete level, and show interesting physics of SOC spin-1 BECs, such as the SOC effects and different wave patterns.