Integrable boundary for quad-graph systems: Three-dimensional boundary consistency
Integrable boundary for quad-graph systems: Three-dimensional boundary consistency
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DOI：
10.3842/sigma.2014.014
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发表时间：
2013-07
影响因子：
0.9
通讯作者：
V. Caudrelier;Nicolas Cramp'e;Qi Zhang
中科院分区：
文献类型：
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作者：
V. Caudrelier;Nicolas Cramp'e;Qi Zhang
在四元图上离散可积系统的背景下，我们提出了可积边界的概念。表征边界的方程必须满足与表征体积的方程的相容性方程，我们称之为三维（3D）边界相容性。与通常的与立方体相关联的3D一致性条件相比，我们的3D边界一致性条件存在于菱形十二面体的一半上。我们提供了一个列表的可积边界相关联的每个四边形方程的分类阿德勒，Bobenko和Suris。然后，使用术语“可积边界”是合理的事实，有Backlund变换和零曲率表示的边界满足我们的条件的系统。我们讨论了边界方程的三边形式，得到了相应的带边界的离散Toda型模型，并作为特例恢复了以前的结果。最后，建立了三维边界一致性与集合论反射方程之间的联系。
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 Backlund 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.