  We study the dynamics of emission of radiation (small-amplitude waves) in
fast collisions between two solitons of the nonlinear Schr\"odinger (NLS)
equation in the presence of weak cubic loss. We calculate the radiation
dynamics by a perturbation technique with two small parameters: the cubic loss
coefficient $\epsilon_{3}$ and the reciprocal of the group velocity difference
$1/\beta$. The agreement between the perturbation theory predictions and the
results of numerical simulations with the full coupled-NLS propagation model is
very good for large $\beta$ values, and is good for intermediate $\beta$
values. Additional numerical simulations with four simplified NLS models show
that the differences between perturbation theory and simulations for
intermediate $\beta$ values are due to the effects of Kerr nonlinearity on
interpulse interaction in the collision. Thus, our study demonstrates that the
perturbation technique that was originally developed to study radiation
dynamics in fast soliton collisions in the presence of conservative
perturbations can also be employed for soliton collisions in the presence of
dissipative perturbations.
