Integrability and asymptotic behaviour of a differential-difference matrix equation

Gordoa, Pilar R.; Pickering, Andrew; Wattis, Jonathan A.D.

Abstract

In this paper we consider the matrix lattice equation U_{n,t} (U_{n+1} ? U_{n?1} ) = g(n)I, in both its autonomous (g(n) = 2) and nonautonomous (g(n) = 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.

• Journal Article Type: Article
• Acceptance Date: Sep 23, 2020
• Online Publication Date: Sep 28, 2020
• Publication Date: 2021-01
• Deposit Date: Oct 6, 2020
• Publicly Available Date: Sep 29, 2021
• Journal: Physica D: Nonlinear Phenomena
• Print ISSN: 0167-2789
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 415
• Article Number: 132754
• DOI: https://doi.org/10.1016/j.physd.2020.132754
• Keywords: Statistical and Nonlinear Physics; Condensed Matter Physics
• Public URL: https://nottingham-repository.worktribe.com/output/4942235
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0167278920303067
• Additional Information: This article is maintained by: Elsevier; Article Title: Integrability and asymptotic behaviour of a differential-difference matrix equation; Journal Title: Physica D: Nonlinear Phenomena; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.physd.2020.132754; Content Type: article; Copyright: © 2020 Published by Elsevier B.V.