Caudrelier, V., Crampe, N. & Zhang, Q. C. (2014). Integrable boundary for quadgraph systems: Threedimensional boundary consistency. SIGMA 10, 14, . doi: 10.3842/SIGMA.2014.014

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Abstract
We propose the notion of integrable boundary in the context of discrete integrable systems on quadgraphs. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called the threedimensional (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 quadgraph 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 threeleg form of boundary equations, obtain associated discrete Todatype models with boundary and recover previous results as particular cases. Finally, the connection between the 3D boundary consistency and the settheoretical reflection equation is established.
Item Type:  Article 

Uncontrolled Keywords:  Discrete integrable systems, quadgraph equations, 3Dconsistency, Bäcklund transformations, zero curvature representation, Todatype systems, settheoretical reflection equation 
Subjects:  Q Science > QA Mathematics 
Divisions:  School of Engineering & Mathematical Sciences > Department of Mathematical Science 
Related URLs:  
URI:  http://openaccess.city.ac.uk/id/eprint/3237 
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