Classification of integrable boundary equations for integrable quad-graph systems

TL;DR

This paper classifies integrable boundary equations for quad-graph systems by formalizing boundary conditions and solving boundary consistency, extending the Adler-Bobenko-Suris classification to include boundary considerations.

Contribution

It introduces a systematic method for classifying integrable boundary equations for quad-graph systems, expanding the understanding of boundary conditions in integrable discrete systems.

Findings

Classification of integrable boundary equations for quad-graph systems.

A systematic method based on boundary consistency and factorization.

Extension of the classification to rhombic-symmetric equations.

Abstract

In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was introduced as a criterion for defining integrable boundary conditions for quad-graph systems with a boundary. In this paper, we formalize the notions of boundary equations as boundary conditions for quad-graph systems, and provide a systematic method for solving the boundary consistency, which results in a classification of integrable boundary equations for quad-graph equations in the Adler-Bobenko-Suris classification. This relies on factorizing, first the quad-graph equations into pairs of dual boundary equations, and then the consistency on a rhombic dodecahedron into two equivalent boundary consistencies. Generalizations of the method to rhombic-symmetric equations are also considered.