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Integrable boundary for quad-graph systems: Three-dimensional boundary consistency

Caudrelier, V., Crampe, N. & Zhang, Q. C. (2014). Integrable boundary for quad-graph systems: Three-dimensional boundary consistency. SIGMA 10, 10, -. doi: 10.3842/sigma.2014.014


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.

Publication Type: Article
Publisher Keywords: Discrete integrable systems, quad-graph equations, 3D-consistency, Bäcklund transformations, zero curvature representation, Toda-type systems, set-theoretical reflection equation
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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