In this paper we consider the matrix lattice equation ${\mathbf{U}}_{n,t}({\mathbf{U}}_{n+1}-{\mathbf{U}}_{n-1})=g\left(n\right)\mathbf{I}$, in both its autonomous ($g\left(n\right)=2$) and nonautonomous ($g\left(n\right)=2n-1$) forms. We show that each of these two matrix lattice equations are integrable. In addition, we explore the construction of Miura maps which relate these two lattice equations, via intermediate equations, to matrix analogs of autonomous and nonautonomous Volterra equations, but in two matrix dependent variables. For these last systems, we consider cases where the dependent variables belong to certain special classes of matrices, and obtain integrable coupled systems of autonomous and nonautonomous lattice equations and corresponding Miura maps. Moreover, in the nonautonomous case we present a new integrable nonautonomous matrix Volterra equation, along with its Lax pair. Asymptotic reductions to the matrix potential Korteweg–de Vries and matrix Korteweg–de Vries equations are also given.

在本文中，我们考虑矩阵格方程 ${\mathbf{ü}}_{ñ，Ť}（{\mathbf{ü}}_{ñ+\mathrm{1个}}-{\mathbf{ü}}_{ñ-\mathrm{1个}}）=G（ñ）\mathbf{一世}$，无论是在自主（$G（ñ）=2$）和非自治（$G（ñ）=2ñ-\mathrm{1个}$） 形式。我们证明这两个矩阵晶格方程都是可积的。此外，我们探索构造Miura映射的方法，该映射通过中间方程将这两个晶格方程与自主和非自主Volterra方程的矩阵类似物关联，但是存在两个依赖于矩阵的变量。对于这些最后的系统，我们考虑因变量属于矩阵的某些特殊类的情况，并获得自治和非自治晶格方程和相应的Miura映射的可积耦合系统。此外，在非自治情况下，我们提出了一个新的可积非自治矩阵Volterra方程及其Lax对。还给出了矩阵势Korteweg-de Vries和矩阵Korteweg-de Vries方程的渐近约化。