Classification of integrable discrete equations of octahedron type

TL;DR

This paper classifies integrable discrete 3D equations of octahedron type using the consistency approach, highlighting key examples like the discrete KP and Schwarzian equations, and explaining their geometric and combinatorial structures.

Contribution

It provides a comprehensive classification of octahedron type equations, including their geometric interpretation and the tripodal form as a key technical tool.

Findings

Includes the discrete KP and Schwarzian equations.

Shows the geometric origin from Desargues theorem.

Provides a classification framework for these equations.

Abstract

We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the most prominent representatives of this class, the discrete KP equation and its Schwarzian (multi-ratio) version, as well as three further equations. The combinatorics and geometry of the octahedron type equations are explained. In particular, the consistency on the 4-dimensional Delaunay cells has its origin in the classical Desargues theorem of projective geometry. The main technical tool used for the classification is the so called tripodal form of the octahedron type equations.