Asymptotic expansions and solitons of the Camassa–Holm – nonlinear Schrödinger equation

Abstract

We study a deformation of the defocusing nonlinear Schrödinger (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.