Lax pair, binary Darboux transformation and dark solitons for the three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

In this paper, we investigate the dark solitons for the three-component Gross-Pitaevskii system, which describes the F=1 spinor Bose-Einstein condensate, with F denoting the atom's spin. We construct a Lax pair and a binary Darboux transformation for such a system. Based on a nonzero seed solution, we obtain the kink and dark solitons corresponding to the ferromagnetic and polar states, respectively. Moreover, we obtain the W-shaped dark soliton. Interactions between the two dark solitons are discussed. We find that the interaction is inelastic if a kink soliton is included.