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 Møller 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(ɛx). We prove that the soliton decomposes into a soliton plus a scattering wave (radiation) up to times of order ɛ−1. To leading order, the center of the soliton follows the trajectory of a classical particle in the potential W(ɛx).
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