Stability and dynamics of self-bound state of spin–orbit coupled spin-1 Bose–Einstein condensates

We study self-bound state of spin–orbit (SO) coupled spin-1 BECs under the action of the SO coupling and the density-dependent and spin-dependent interactions. The phase transition conditions from the magnetized phase to the unmagnetized phase are analytically obtained in the self-bound state, and the physical mechanism of the phase transition is revealed. The distinct properties of self-bound state in different ground states are discussed in detail, and static and moving self-bound states are obtained. Then, the stability of the self-bound state is studied, and the unstable threshold caused by the competition among SO coupling, Raman coupling, spin-dependent interaction and density-dependent interaction is obtained analytically and confirmed numerically. Furthermore, the dynamical transition of self-bound state from moving type to static type is studied numerically. Due to the spin momentum locking and the drive of spin dynamics, the self-bound states exhibit distinct dynamical behaviors in different ground state phases. Our study provides theoretical evidence for a deep understanding of the stability and the dynamic of self-bound states in SO coupled spin-1 BECs.