  The interaction of matter-wave solitons with a potential barrier is a
fundamentally important problem, and the splitting and subsequent recombination
of the soliton by the barrier is the essence of soliton matter-wave
interferometry. We demonstrate the three-dimensional (3D) character of the
interactions in the case relevant to ongoing experiments, where the number of
atoms in the soliton is sufficiently large that the soliton is close to
collapse. The mean-field description is quite accurate, but the proximity to
the collapse threshold makes the use of the 1D Gross-Pitaevskii equation (GPE)
irrelevant. We examine the soliton dynamics in the framework of the effectively
1D nonpolynomial Schr{\"{o}}dinger equation (NPSE), which admits the collapse
in a modified form, and in parallel we use the full 3D GPE. Both approaches
produce results which are very different from those produced in recent work
which used the 1D cubic GPE. The features observed in both the NPSE and the 3D
GPE include (a) an increase in the first reflection coefficient for increasing
barrier height and decreasing atom number; (b) strong modulation in the
secondary reflection/recombination probability vs. barrier height; (c) a
pronounced asymmetry in the oscillation amplitudes of the transmitted and
reflected fragments; (d) an enhancement of the transverse excitations as the
number of atoms is increased. We also explore effects produced by variations of
the barrier width and outcomes of the secondary collision upon phase imprinting
on the fragment in one arm of the interferometer.
