标题： 达布变换及相关非阿贝尔可积微分差分系统的一类导数非线性薛定谔型 显示英文标题

标题： Darboux transformations and related non-Abelian integrable differential-difference systems of the derivative nonlinear Schrödinger type

摘要： 我们构建了与Lax算子的约化群相容的线性和二次Darboux矩阵，适用于每个已知的具有Lax表示的七种非阿贝尔导数非线性Schrödinger方程。 从这些Darboux变换中推导出的微分差分系统通过引入非交换常数来推广已建立的非阿贝尔可积模型。 具体而言，我们证明线性Darboux变换生成非阿贝尔Volterra型方程，而二次变换产生双组分系统，包括非阿贝尔形式的Ablowitz-Ladik、Merola-Ragnisco-Tu和相对论Toda方程。 利用准行列式，我们建立了将高次多项式Darboux矩阵分解为特定线性Darboux矩阵作为因子的必要条件。 这一结果使得二次Darboux矩阵可以分解为一对线性Darboux矩阵。 显示英文摘要

摘要： 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.