Darboux transformations, finite reduction groups and related Yang–Baxter maps

Abstract

In this paper, we construct Yang–Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present six-dimensional YB maps corresponding to Darboux transformations for the nonlinear Schrödinger (NLS) equation and the derivative nonlinear Schrödinger (DNLS) equation. These YB maps can be restricted to four-dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries (Fordy A and Kassotakis A 2013 J. Phys. A: Math. Theor. 46 205201). We give a six-dimensional YB-map corresponding to the Darboux transformation for a deformation of the DNLS equation. We also consider vector generalizations of the YB maps corresponding to the NLS and DNLS equations.