  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 Schroedinger 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 as well as 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.
