Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlev\'{e} equations

TL;DR

This paper introduces noncommutative nonisospectral Toda and Lotka-Volterra lattices, deriving matrix discrete Painlevé equations and providing quasideterminant solutions, expanding the understanding of integrable systems in noncommutative settings.

Contribution

It presents the first noncommutative nonisospectral lattices and derives associated matrix Painlevé equations with explicit quasideterminant solutions.

Findings

Derived matrix discrete Painlevé I equations from noncommutative lattices.

Established quasideterminant solutions for these Painlevé equations.

Connected stationary reductions to rational solutions in the noncommutative context.

Abstract

The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without specific weight functions, respectively. Under stationary reductions, matrix discrete Painlev\'{e} I and matrix asymmetric discrete Painlev\'{e} I equations are derived separately not only from the noncommutative nonisospectral lattices themselves, but also from their Lax pairs. The rationality of the stationary reduction has been justified in the sense that quasideterminant solutions are provided for the corresponding matrix discrete Painlev\'{e} equations.