Unexpected behavior of nonlinear Schrödinger solitons in an external potential.

Employing a higher order symplectic integration algorithm, we integrate the 1-d nonlinear Schrr odinger equation numerically for solitons moving in an external potential. In particular, we study the scattering oo an interface separating two regions of constant potential, with a linearly rising ramp in the interface region. Transmission coeecients are computed as functions of the height of this ramp and of its steepness in the semi-classical domain (slope 1) near the classical threshold for transmission. For energies slightly above this threshold, we nd a narrow range of slopes in which the soliton is almost fully reeected, though it is nearly completely transmitted outside this range. Recent years have seen a considerable growth in the interest for nonlinear partial diierential equations with soliton solutions. In particular, the nonlinear Schrr odinger equation (NLSE) and its modiications have a quite large range of applicability, describing phenomena occurring in optics, solid state, particle and plasma physics. Some of the more interesting problems, for which the NLSE appears to provide a fruitful description, are modulational instability of water waves 1], propagation of heat pulses in anharmonic crystals, helical motion of very thin vortex laments, nonlinear modulation of collisionless plasma waves 2], and self-trapping of light beams in optically nonlinear media 3, 4, 5]. In all these problems, the main interest lies in the fact that the NLSE has soliton solutions. These are solitary waves with well deened pulse-like shapes and remarkable stability properties 6]. A great deal of current interest is directed to the innuence of external perturbations on these states. Such perturbations can be due to diierent sources. One interesting class consists of \Hamiltonian" perturbations which do not destroy the Hamiltonian structure of the NLSE. They are essentially of two kinds, one arising from the interaction with a potential term, the other due to spatial inhomogeneity 1