On the point-particle (Newtonian) limit of the non-linear Hartree equation

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear Moller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential W(epsilonx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) Lip to times of order epsilon(-1). To leading order, the center of the soliton follows the trajectory of a classical particle in the potential W(epsilonx).