  We derive a closed expression for the spatial shift experienced by a black
soliton colliding with a shallow dark soliton in the context of deformed
non-linear Schr\"{o}dinger models. A perturbative scheme is developed based on
the expansion parameter $1/(\g v)<<1$, where $v$ is the velocity of the
incoming shallow dark soliton, $\g \equiv 1/\sqrt{1-v^2/v_s^2}$\, and $v_s$ is
the Bogoliubov sound speed, therefore it is not restricted to small
deformations of the integrable NLS model. As applications of our formalism we
consider the integrable NLS model and the non-integrable cubic-quintic NLS
model with non-vanishing boundary conditions. Extensive numerical simulations
are performed in order to verify our results. A variant of the analysis for
gray-gray soliton collision is discussed regarding a fast broad soliton and a
slow thin soliton.
