Darboux transformations and related non-Abelian integrable differential-difference systems of the derivative nonlinear Schr\"odinger type

TL;DR

This paper develops Darboux transformations for non-Abelian derivative nonlinear Schrödinger equations, leading to new integrable differential-difference systems and factorisation methods using quasideterminants.

Contribution

It introduces linear and quadratic Darboux matrices for non-Abelian DNLS equations, generalising known models with non-commutative constants and establishing factorisation conditions.

Findings

Linear Darboux transformations generate non-Abelian Volterra-type equations.

Quadratic Darboux transformations produce two-component non-Abelian systems.

Quasideterminants are used to factorise higher-degree Darboux matrices.

Abstract

We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schr\"odinger equations that admit Lax representations. The differential-difference systems derived from these Darboux transformations generalise established non-Abelian integrable models by incorporating non-commutative constants. Specifically, we demonstrate that linear Darboux transformations generate non-Abelian Volterra-type equations, while quadratic transformations yield two-component systems, including non-Abelian versions of the Ablowitz-Ladik, Merola-Ragnisco-Tu, and relativistic Toda equations. Using quasideterminants, we establish necessary conditions for factorising a higher-degree polynomial Darboux matrix with a specific linear Darboux matrix as a factor. This result enables the factorisation of quadratic Darboux matrices into pairs of linear Darboux matrices.