Non-invertible transformations for the classification of differential-difference equations

TL;DR

This paper explores non-invertible transformations in classifying differential-difference equations, introducing new concepts and techniques, and illustrating them with a novel integrable equation linked to known models.

Contribution

It introduces the concept of non-Miura type linearizable transformations and provides methods to construct and utilize them for classification.

Findings

Developed techniques for constructing linearizable transformations.

Identified a new integrable differential-difference equation.

Connected the new equation to the Itoh–Narita–Bogoyavlensky equation.

Abstract

We discuss aspects of the theory of non-invertible transformations which enter in the problem of classification of diffe\-ren\-tial-difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non--Miura type linearizable transformation and we present techniques which allow one to construct simple linearizable transformations and help us to solve the classification problem. This theory is illustrated by the example of a new integrable differential--difference equation depending on 5 lattice points, interesting from the viewpoint of the non-invertible transformation which relate it to an Itoh--Narita--Bogoyavlensky equation.