### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 014, 24 pages      arXiv:1307.4023      https://doi.org/10.3842/SIGMA.2014.014

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

Vincent Caudrelier a, Nicolas Crampé b and Qi Cheng Zhang a
a) Department of Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK
b) CNRS, Laboratoire Charles Coulomb, UMR 5221, Place Eugène Bataillon - CC070, F-34095 Montpellier, France

Received July 19, 2013, in final form February 05, 2014; Published online February 12, 2014

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 Bäcklund 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.

Key words: discrete integrable systems; quad-graph equations; 3D-consistency; Bäcklund transformations; zero curvature representation; Toda-type systems; set-theoretical reflection equation.

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