Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency

TL;DR

This paper introduces the concept of integrable boundaries for quad-graph systems, establishing a new 3D boundary consistency condition on a rhombic dodecahedron, and explores their properties and connections to other integrable structures.

Contribution

It defines a novel 3D boundary consistency condition for discrete integrable systems on quad-graphs and provides classifications, transformations, and links to reflection equations.

Findings

List of integrable boundaries for classified quad-graph equations

Existence of Bäcklund transformations and zero curvature representations with boundary

Connection to set-theoretical reflection equation

Abstract

We propose the notion of integrable boundary in the context of discrete integrable systems on quad-graphs. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called the three-dimensional (3D) boundary consistency. In comparison to the usual 3D consistency condition which is linked to a cube, our 3D boundary consistency condition lives on a half of a rhombic dodecahedron. The We provide a list of integrable boundaries associated to each quad-graph equation of the classification obtained by Adler, Bobenko and Suris. Then, the use of the term "integrable boundary" is justified by the facts that there are B\"acklund transformations and a zero curvature representation for systems with boundary satisfying our condition. We discuss the three-leg form of boundary equations, obtain associated discrete Toda-type models with boundary and recover previous results as particular cases. Finally, the connection between the 3D boundary consistency and the set-theoretical reflection equation is established.