Resonant trapping in the transport of a matter-wave soliton through a quantum well

We theoretically investigate the scattering of bright solitons in a Bose-Einstein condensate on narrow attractive potential wells. Reflection, transmission, and trapping of an incident soliton are predicted to occur with remarkably abrupt transitions upon varying the potential depth. Numerical simulations of the nonlinear Schr\"odinger equation are complemented by a variational collective coordinate approach. The mechanism for nonlinear trapping is found to rely both on resonant interaction between the soliton and bound states in the potential well and on the radiation of small-amplitude waves. These results suggest that solitons can be used to probe bound states that are not accessible through scattering with single atoms.