Dark matter-wave bound soliton molecules and modulation stability for spin-1 Gross–Pitaevskii equations in Bose–Einstein condensates

The non-autonomous dark soliton solutions of the spin-1 Gross–Pitaevskii equations with a parabola external potential are considered. The spin-1 Gross–Pitaevskii equations are transformed into the constant-coefficient (autonomous) three-component Gross–Pitaevskii (3-GP) equations by using the similarity transformation method, the dark soliton solutions of the autonomous 3-GP equations are generated by using the Hirota’s bilinear method. Then we investigate the modulation stability (MS) of the autonomous 3-GP equations and find the relation between Ω and κ. The regions of MS and modulation instability (MI) are distinguished, and the stable solitons from the MS region are derived by selecting specific parameters. Since the bright soliton solutions of the spin-1 Gross–Pitaevskii equations have been described in detail in the other literatures, here we focus on the dark soliton solutions. We obtain non-autonomous dark one-soliton, dark two-soliton and some novel forms of dark soliton solutions, such as “X-typed”, “Y-typed”, “S-typed” and “Ψ-typed” wave solutions and some soliton interactions of the spin-1 Gross–Pitaevskii equations. Some special dynamic behaviors of dark solitons of the spin-1 Gross–Pitaevskii equations with an external potential are obtained via selecting some free functions. These obtained non-autonomous dark soliton solutions can give some potential applications in Bose–Einstein condensates (BECs).