  The effective long-time dynamics of solitary wave solutions of the nonlinear
Schr\"odinger equation in the presence of rough nonlinear perturbations is
rigorously studied. It is shown that, if the initial state is close to a slowly
travelling soliton of the unperturbed NLS equation (in $H^1$ norm), then, over
a long time scale, the true solution of the initial value problem will be close
to a soliton whose center of mass dynamics is approximately determined by an
effective potential that corresponds to the restriction of the nonlinear
perturbation to the soliton manifold.
