Integrability and asymptotic behaviour of a differential-difference matrix equation

Highlights: • We give Miura maps for autonomous and nonautonomous matrix lattice equations and systems. • We prove the integrability of an autonomous and also of a nonautonomous matrix lattice. • We obtain integrable multicomponent autonomous and nonautonomous lattices and Miura maps. • We give a new integrable nonautonomous matrix Volterra equation along with its Lax pair. • We give asymptotic reductions of matrix lattices to matrix potential KdV and matrix KdV. In this paper we consider the matrix lattice equation $U_{n,t}(U_{n+1}-U_{n-1})=g\left(n\right)I$, 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.