# $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems and Darboux transformations

## TL;DR

This paper develops Darboux transformations for a new class of two-dimensional discrete integrable systems called $
Z_N$ graded systems, derived from the Gel'fand-Dikii hierarchy, revealing a unified solution structure.

## Contribution

It introduces Darboux transformations for $
Z_N$ graded discrete integrable systems and derives their Lax pairs from the Gel'fand-Dikii hierarchy, expanding the understanding of their solution structures.

## Key findings

- Derived $
Z_N$ graded discrete equations and Lax pairs from the hierarchy.
- Constructed Darboux transformations using linear problems in bilinear formalism.
- Unified solution structure for all $
Z_N$ graded equations.

## Abstract

We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the framework of $\mathbb{Z}_\mathcal{N}$ graded discrete Lax pairs very recently. In this paper, the $\mathbb{Z}_\mathcal{N}$ graded discrete equations in coprime case and their corresponding Lax pairs are derived from the discrete Gel'fand-Dikii hierarchy by applying a transformation of the independent variables. The construction of the Darboux tranformations is realised by considering the associated linear problems in the bilinear formalism for the $\mathbb{Z}_\mathcal{N}$ graded lattice equations. We show that all these $\mathbb{Z}_\mathcal{N}$ graded equations share a unified solution structure in our scheme.