  We integrate the one-dimensional nonlinear Schr\"odinger equation numerically
for solitons moving in external potentials. In particular, we study the
scattering off an interface separating two regions of constant potential
modeled by a linear ramp. Transmission coefficients and inelasticities are
computed as functions of the potential difference and the slope of the ramp.
Our data show that the ramp's slope has a strong influence revealing unexpected
windows of reflection in a transmission regime. The transmission coefficients
for very small potential differences are compared with the theoretical
predictions derived by perturbation theory. Also the time evolution of the
solitary waves after the scattering is studied. We observed that they in
general behave like solitons with an amplified amplitude. Due to this, they
oscillate. The oscillation period is measured and compared with theoretical
predictions.
  PACS number(s): 03.40.Kf, 0.3.65.Ge, 42.65.Tg, 42.81.Dp
