Dynamics and stability of stationary states for the spin-1 Bose–Einstein condensates in a standing light wave

• Stationary solutions of spin-1 BECs with periodic potential are proposed.
• Stability is analyzed by linear stability analysis and dynamical evolutions.
• Antiferromagnetic condensates are more stable than ferromagnetic ones.

The spin-1 Bose–Einstein condensates trapped in a standing light wave can be described by three coupled Gross–Pitaevskii equations with a periodic potential. In this paper, nine families of stationary solutions without phase structures in the form of Jacobi elliptic functions are proposed, and their stabilities are analyzed by both linear stability analysis and dynamical evolutions. Taking the ferromagnetic 87Rb atoms and antiferromagnetic (polar) 23Na atoms as examples, we investigate the stability regions of the nine stationary solutions, which are given in term of elliptic modulus k. It is shown that for the same stationary solution the stability regions of condensates with antiferromagnetic (polar) spin-dependent interactions are larger than that of the condensates with ferromagnetic ones. The dn-dn-dn stationary solution is the most stable solution among the nine families of stationary solutions. Moreover, in the same standing light wave, the spin-1 Bose–Einstein condensates are more stable than the scalar Bose–Einstein condensate.