  We study the collision of two fast solitons for the nonlinear Schr\"odinger
equation in the presence of a spatially adiabatic external potential. For a
high initial relative speed $\|v\|$ of the solitons, we show that, up to times
of order $\log\|v\|$ after the collision, the solitons preserve their shape (in
$L^2$-norm), and the dynamics of the centers of mass of the solitons is
approximately determined by the external potential, plus error terms due to
radiation damping and the extended nature of the solitons. We remark on how to
obtain longer time scales under stronger assumptions on the initial condition
and the external potential.
