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arXiv · 1906.11593

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

Ying Shi
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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.

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Ying Shi. 2019-06-27. $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems and Darboux transformations. https://doi.org/10.1088/1751-8121%2Fab59b3

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