Solitary Waves of the Camassa-Holm Derivative Nonlinear Schr\"odinger equation

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.