Caudrelier, V, Crampé, N and Zhang, QC (2014) Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency. SIGMA, 10. 014. p. 24. ISSN 1815-0659
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.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | This is an open access article under the terms of the Creative Commons Attribution-ShareAlike License [https://creativecommons.org/licenses/by-sa/4.0/] |
Keywords: | nlin.SI; discrete integrable systems; quad-graph equations; 3D-consistency; Bäcklund transformations; zero curvature representation; Toda-type systems; set-theoretical reflection equation. |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 08 Sep 2017 12:19 |
Last Modified: | 08 Sep 2017 12:19 |
Published Version: | https://doi.org/10.3842/SIGMA.2014.014 |
Status: | Published |
Publisher: | SIGMA |
Identification Number: | 10.3842/SIGMA.2014.014 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:109001 |