Unexpected behavior of nonlinear Schrödinger solitons in an external potential

Employing a higher order symplectic integration algorithm, we integrate the one-dimensional nonlinear Schr\"odinger equation numerically for solitons moving in an external potential. In particular, we study the scattering off an interface separating two regions of constant potential, with a linearly rising ramp in the interface region. Transmission coefficients are computed as functions of the height of this ramp and of its steepness in the semiclassical domain (slope{}1) near the classical threshold for transmission. For energies slightly above this threshold, we find a narrow range of slopes in which the soliton is almost fully reflected, though it is nearly completely transmitted outside this range. {} 1996 The American Physical Society.