In this paper we consider the scattering of a soliton or solitary wave
by a linear potential. By careful treatment of the radiation we show
that the amount of mass and energy lost by the solitary wave during a
scattering event is exponentially small for strong nonlinearities. The
constant associated with this exponentially small radiation is expres
sed in terms of the binding energy of the soliton (solitary wave), and
the analyticity properties of the potential and the soliton (solitary
wave). This calculation does not use integrability in any way. In the
case of a delta function potential and the cubic NLS, our results agr
ee with the more explicit results derived by Kivshar, Gredeskul, Sanch
ez, and Vasquez using perturbation theory based on the inverse scatter
ing transform. Following them, we take the limit of a continuum of wel
l separated scatterers, and derive a closed system of ordinary differe
ntial equations. Analyzing the limiting behavior of these equations fo
r large distance Z into the medium we find that the velocity of the so
liton decays as (log(Z))(-1) for a delta function potential or a poten
tial which is meromorphic as a function of a complex variable, and mor
e slowly than (log(Z))(-1) for a potential which is an entire function
of a complex variable.