Classification of 3D consistent quad-equations

TL;DR

This paper classifies 3D consistent systems of quad-equations on a cube, extending known integrable equations to lattice structures and providing Bäcklund transformations and zero-curvature representations.

Contribution

It extends the ABS classification to lattice systems, demonstrating integrability through Bäcklund transformations and zero-curvature representations.

Findings

Includes the ABS-list within the classification

Provides at least one integrable system per quad-equation

Establishes Bäcklund transformations and zero-curvature representations

Abstract

We consider 3D consistent systems of six independent quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice Z^3 is possible by reflecting the cubes. For every quad-equation we will give at least one system included leading to a B\"acklund transformation and a zero-curvature representation which means that they are integrable.