Slow soliton interaction with delta impurities

We study the Gross--Pitaevskii equation with a delta function potential, $ q \delta_0 $, where $ |q| $ is small and analyze the solutions for which the initial condition is a soliton with initial velocity $ v_0 $. We show that up to time $ (|q| + v_0^2 )^{-1/2} \log$($1$ / $|q|$) the bulk of the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (\xi^2 + q \, \sech^2 ( x ) )$ / $2$.