Multiple Soliton Asymptotics in a Spin-1 Bose–Einstein Condensate

Spinor Bose–Einstein condensates (BECs) are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom. This study concentrates on a (1+1)-dimensional three-couple Gross–Pitaevskii system to depict the macroscopic spinor BEC waves within the mean-field approximation. Regarding the distribution of the atoms corresponding to the three vertical spin projections, a known binary Darboux transformation is utilized to derive the N matter-wave soliton solutions and triple-pole matter-wave soliton solutions on the zero background, where N is a positive integer. For those multiple matter-wave solitons, the asymptotic analysis is performed to obtain the algebraic expressions of the soliton components in the N matter-wave solitons and triple-pole matter-wave solitons. The asymptotic results indicate that the matter-wave solitons in the spinor BECs possess the property of maintaining their energy content and coherence during the propagation and interactions. Particularly, in the N matter-wave solitons, each soliton component contributes to the phase shifts of the other soliton components; and in the triple-pole matter-wave solitons, stable attractive forces exist between the different matter-wave soliton components. Those multiple matter-wave solitons are graphically illustrated through three-dimensional plots, density plot and contour plot, which are consistent with the asymptotic analysis results. The present analysis may provide the explanations for the complex natural mechanisms of the matter waves in the spinor BECs, and may have potential applications in designs of atom lasers, atom interferometry and coherent atom transport.