Symmetries of Integrable Difference Equations on the Quad‐Graph

Abstract

This paper describes symmetries of all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

“…such that α, β are constants. We present here some results on the non-autonomous case when α and β depend on n and m. Similar results can be found in [24].…”

“…Moreover we prove that the symmetries of the ABS equations [24,28] are subcases of the YdKN equation. For the sake of clarity we consider in a more detailed way just the case of equation (H3).…”

“…In the papers [24,28] the three and five-point generalized symmetries have been found for all equations of the ABS list. We shall use these results to show that the ABS equations may be interpreted as Bäcklund transformations for the differential-difference YdKN equation [30].…”

“…In [24,28] generalized symmetries have been obtained for autonomous ABS equations, i.e. such that α, β are constants.…”

“…We refer to the papers [3,24,28,17,18,27] for some recents results about these equations. Our main purpose is the analysis of their transformation properties.…”