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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2014
• Main Authors: Caudrelier, V., Crampé, N., Zhang, Q.C.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2014
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/146841
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency / V. Caudrelier, N. Crampé, Q.C. Zhang // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 30 назв. — англ.

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.