  We investigate scattering bright solitons off a potential using both
analytical and numerical methods. Our paper focuses on low kinetic energies for
which differences between the mean-field description via the Gross-Pitaevskii
equation (GPE) and the quantum behavior are particularly large. On the
N-particle quantum level, adding an additional harmonic confinement leads to a
simple signature to distinguish quantum superpositions from statistical
mixtures. While the non-linear character of the GPE does not allow quantum
superpositions, the splitting of GPE-solitons takes place only partially. When
the potential strength is increased, the fraction of the soliton which is
transmitted or reflected jumps non-continuously. We explain these jumps via
energy-conservation and interpret them as indications for quantum
superpositions on the N-particle level. On the GPE-level, we also investigate
the transition from this stepwise behavior to the continuous case.
