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Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10
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Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
Інший ID:
2010 Mathematics Subject Classification: 05C10; 37K10; 39A12; 57M15
DOI:10.3842/SIGMA.2014.014
DOI:10.3842/SIGMA.2014.014
Посилання:
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency / V. Caudrelier, N. Crampé, Q.C. Zhang // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 30 назв. — англ.
Підтримка:
The final details of this paper were completed while two of the authors (V.C. and Q.C.Z) were
at the “Discrete Integrable Systems” conference held at the Newton Institute for Mathematical
Sciences. We wish to thank C. Viallet for pointing out useful references. We also thank
M. Nieszporski and P. Kassotakis for useful discussions and the provision of unpublished material
on their work on the connection between Yang–Baxter maps and quad-graph equations, some
of which is related to our results shown in Table 3. Last, but not least, we express our sincere
gratitude to the referees whose excellent comments and criticisms helped improve this paper
tremendously.
Дата:
2014
Переглядів:
377
Завантажень:
252
Анотація:
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.
