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

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.