  In this paper we examine a deformation of the derivative nonlinear
Schr\"odinger (DNLS) equation, the so-called Camassa-Holm DNLS (CH-DNLS)
equation. We use two asymptotic multiscale expansion methods to reduce this
model to both the modified Korteweg-de Vries (MKdV) equation and the
Korteweg-de Vries(KdV) equation. Using their exact soliton solutions, we
construct approximate solutions of the original CH-DNLS equation, namely dark
and anti-dark solitary waves. Direct numerical simulations demonstrate the
validity of these approximate solutions and illustrate their dynamical
evolution, including their potential for nearly elastic head-on collisions in
the case examples considered.
