  Reflection of wave packets from downward potential steps and attractive
potentials, known as a quantum reflection, has been explored for bright
matter-wave solitons with the main emphasis on the possibility to trap them on
top of a pedestal-shaped potential. In numerical simulations, we observed that
moving solitons return from the borders of the potential and remain trapped for
a sufficiently long time. The shuttle motion of the soliton is accompanied by
shedding some amount of matter at each reflection from the borders of the trap,
thus reducing its norm. The one- and two- soliton configurations are
considered. A discontinuous jump of trajectories of colliding solitons has been
discussed. The time-shift observed in a step-like decay of the moving soliton's
norm in the two-soliton configuration is linked to the trajectory jump
phenomenon. The obtained results can be of interest for the design of new
soliton experiments with Bose-Einstein condensates.
