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

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.

22 pages

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