Non-invertible transformations of differential–difference equations

Abstract

We discuss aspects of the theory of non-invertible transformations of differential–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 that allow one to construct simple linearizable transformations and might help one to solve classification problems. This theory is illustrated by the example of a new integrable differential–difference equation depending on five lattice points, interesting from the viewpoint of the non-invertible transformation, which relate it to an Itoh–Narita–Bogoyavlensky equation.