Vector solitons for the coupled Gross-Pitaevskii equations with a time-dependent external harmonic potential in the Bose-Einstein condensation

In this paper, we study the nonautonomous vector solitons for the coupled Gross-Pitaevskii equations with a time-dependent external harmonic potential, which can be used to describe the properties of the two-component Bose-Einstein condensation in two hyperfine states. Based on the Kadomtsev-Petviashvili hierarchy reduction, we construct the dark-dark, bright-dark and bright-bright soliton solutions in terms of the Gramian. When the atomic scaled s-wave scattering length a>0, which indicates that the interparticle interaction is repulsive, the N-dark-dark and N-bright-dark solitons exist; when a<0, which indicates that the interparticle interaction is attractive, the N-bright-bright and N-bright-dark solitons exist. With the help of the analytic and graphic analysis, interactions between the two solitons with different forms of the external harmonic potential are studied. Interaction between the two dark-dark solitons is elastic. However, interactions between the two bright-dark solitons and between the two bright-bright solitons may be inelastic under certain parameter constraints. Two bright-dark-soliton and two bright-bright-soliton bound states with varying amplitudes, velocities and widths are presented.