Multi-Component Extension of CAC Systems

TL;DR

This paper introduces a method to generate multi-component integrable systems from scalar quadrilateral systems using cyclic groups, enabling the derivation of higher order equations and reductions with preserved integrability features.

Contribution

It presents a novel approach to construct multi-component integrable lattice systems from scalar systems, extending to higher dimensions and complex reductions.

Findings

Derived multi-component systems inherit integrability features

Constructed higher order lattice equations and Painlevé equations

Extended the approach to higher-dimensional lattice systems

Abstract

In this paper an approach to generate multi-dimensionally consistent $N$-component systems is proposed. The approach starts from scalar multi-dimensionally consistent quadrilateral systems and makes use of the cyclic group. The obtained $N$-component systems inherit integrable features such as B\"acklund transformations and Lax pairs, and exhibit interesting aspects, such as nonlocal reductions. Higher order single component lattice equations (on larger stencils) and multi-component discrete Painlev\'e equations can also be derived in the context, and the approach extends to $N$-component generalizations of higher dimensional lattice equations.