Solitons and modulation instability for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensate

Under investigation in this paper are the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensate. We derive the one- and two-soliton solutions for the one, two and three components of the spinor Bose-Einstein condensate. We also discuss the collisions between the two solitons for one and two components of the spinor Bose-Einstein condensate, and find that the collisions are elastic. We make the linear-stability analysis of the solitons for the one and two components of the spinor Bose-Einstein condensate and obtain the conditions for the modulational instability of the solitons, respectively. Based on the one-soliton solutions for the three components of the spinor Bose-Einstein condensate, we see that the period of the one soliton depends on the frequency in space coordinates x axis, ωx, Planck constant, ℏ, and s-wave scattering lengths of the total spin f = 2, a2. The smaller ωx and ℏ, the longer the period; the larger a2, the longer the period.