# Direct linearisation approach to discrete integrable systems associated with $\mathbb{Z}_\mathcal{N}$ graded Lax pairs

## TL;DR

This paper links Fordy-Xenitidis discrete integrable systems to linear integral equations, revealing their solution structure and connecting them to the discrete Gel'fand-Dikii hierarchy, while also presenting their bilinear form and tau functions.

## Contribution

It establishes a novel connection between Fordy-Xenitidis systems and linear integral equations, providing explicit solutions and bilinear forms.

## Key findings

- Solution structure of Fordy-Xenitidis systems revealed
- Connection to discrete Gel'fand-Dikii hierarchy established
- Bilinear form and tau functions derived

## Abstract

Fordy and Xenitidis [J. Phys. A: Math. Theor. 50 (2017) 165205] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of $\mathbb{Z}_\mathcal{N}$ graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy-Xenitidis discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the Fordy-Xenitidis integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the Fordy-Xenitidis novel models and the discrete Gel'fand-Dikii hierarchy.