  We study a deformation of the defocusing nonlinear Schr\"odinger (NLS)
equation, the defocusing Camassa- Holm NLS, hereafter referred to as CH-NLS
equation. We use asymptotic multiscale expansion methods to reduce this model
to a Boussinesq-like equation, which is then subsequently approximated by two
Korteweg-de Vries (KdV) equations for left- and right-traveling waves. We use
the soliton solution of the KdV equation to construct approximate solutions of
the CH-NLS system. It is shown that these solutions may have the form of either
dark or antidark solitons, namely dips or humps on top of a stable
continuous-wave background. We also use numerical simulations to investigate
the validity of the asymptotic solutions, study their evolution, and their
head-on collisions. It is shown that small-amplitude dark and antidark solitons
undergo quasi-elastic collisions.
