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

Symmetry Integrability and Geometry-Methods and Applications

article

Vincent Caudrelier¹  Nicolas Crampé²  Qi Cheng Zhang¹ 

[1] Department of Mathematical Science, City University London;Laboratoire Charles Coulomb

关键词: discrete integrable systems;    quad-graph equations;    3D-consistency;    B¨acklund transformations;    zero curvature representation;    Toda-type systems;    set-theoretical reflection equation;   

DOI  :  10.3842/SIGMA.2014.014

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown