Prolongation structures and matter-wave solitons in F=1F=1 spinor Bose–Einstein condensate with time-dependent atomic scattering lengths in an expulsive harmonic potential

•Integrability classification of the three component Gross–Pitaevskii equation is proposed.•New nonautonomous matter-wave solitons with varying amplitudes and speeds are obtained.•Collisions of two matter-wave solitons are analyzed and the shape changing interactions are found.

In this paper, the integrability and matter-wave solitons in a system of three component Gross–Pitaevskii equation arising from the context of F=1F=1 spinor Bose–Einstein condensate with time-dependent atomic scattering lengths in an expulsive harmonic potential are investigated by similarity transformation, prolongation technique and Riemann–Hilbert formulation. As a result, some new exact nonautonomous matter-wave soliton solutions with varying amplitudes and speeds are obtained. It is shown that there exist two integrable systems and exact N-matter-wave solitons in spin-1 Bose–Einstein condensates with time-dependent s-wave scattering lengths. The collision dynamics of the two matter-wave solitons are analyzed and the shape changing interaction phenomena are found.