Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
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...
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| Дата: | 2014 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2014
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146841 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-146841 |
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irk-123456789-1468412019-02-12T01:23:11Z Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency Caudrelier, V. Crampé, N. Zhang, Q.C. 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. 2014 Article 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. 1815-0659 2010 Mathematics Subject Classification: 05C10; 37K10; 39A12; 57M15 DOI:10.3842/SIGMA.2014.014 http://dspace.nbuv.gov.ua/handle/123456789/146841 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
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. |
| format |
Article |
| author |
Caudrelier, V. Crampé, N. Zhang, Q.C. |
| spellingShingle |
Caudrelier, V. Crampé, N. Zhang, Q.C. Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Caudrelier, V. Crampé, N. Zhang, Q.C. |
| author_sort |
Caudrelier, V. |
| title |
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency |
| title_short |
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency |
| title_full |
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency |
| title_fullStr |
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency |
| title_full_unstemmed |
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency |
| title_sort |
integrable boundary for quad-graph systems: three-dimensional boundary consistency |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146841 |
| citation_txt |
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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT caudrelierv integrableboundaryforquadgraphsystemsthreedimensionalboundaryconsistency AT crampen integrableboundaryforquadgraphsystemsthreedimensionalboundaryconsistency AT zhangqc integrableboundaryforquadgraphsystemsthreedimensionalboundaryconsistency |
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2023-05-20T17:25:46Z |
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2023-05-20T17:25:46Z |
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1796153272190369792 |