  We study the stability and dynamics of solitons in the Korteweg de-Vries
(KdV) equation with small multiplicative forcing. Forcing breaks the
conservative structure of the KdV equation, leading to substantial changes in
energy over long time. We show that, for small forcing, the inserted energy is
almost fully absorbed by the soliton, resulting in a drastically changed
amplitude and velocity. We decompose the solution to the forced equation into a
modulated soliton and an infinite dimensional perturbation. Assuming slow
exponential decay of the forcing, we show that the perturbation decays at the
same exponential rate in a weighted Sobolev norm centered around the soliton.
