Geodesic
Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: discrete integrable systems; quad-graph equations; 3D-consistency; Bäcklund transformations; zero curvature representation; Toda-type systems; set-theoretical reflection equation. 
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     author = {Vincent Caudrelier and Nicolas Cramp\'e and Qi Cheng Zhang},
     title = {Integrable {Boundary} for {Quad-Graph} {Systems:} {Three-Dimensional} {Boundary} {Consistency}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a13/}
}
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Vincent Caudrelier; Nicolas Crampé; Qi Cheng Zhang. Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a13/

[1] Adler V. E., “Discrete equations on planar graphs”, J. Phys. A: Math. Gen., 34 (2001), 10453–10460 | DOI | MR | Zbl

[2] Adler V. E., Bobenko A. I., Suris Yu. B., “Classification of integrable equations on quad-graphs. {T}he consistency approach”, Comm. Math. Phys., 233 (2003), 513–543, arXiv: nlin.SI/0202024 | MR | Zbl

[3] Adler V. E., Bobenko A. I., Suris Yu. B., “Geometry of {Y}ang–{B}axter maps: pencils of conics and quadrirational mappings”, Comm. Anal. Geom., 12 (2004), 967–1007, arXiv: math.QA/0307009 | DOI | MR | Zbl

[4] Ahn C., Koo W. M., “Boundary {Y}ang–{B}axter in the {RSOS}/{SOS} representation”, Statistical Models, {Y}ang–{B}axter Equation and Related Topics, and {S}ymmetry, Statistical Mechanical Models and Applications ({T}ianjin, 1995), World Sci. Publ., River Edge, NJ, 1996, 3–12, arXiv: hep-th/9508080 | MR

[5] Atkinson J., Hietarinta J., Nijhoff F., “Seed and soliton solutions for {A}dler's lattice equation”, J. Phys. A: Math. Theor., 40 (2007), F1–F8, arXiv: nlin.SI/0609044 | DOI | MR | Zbl

[6] Atkinson J., Joshi N., “Singular-boundary reductions of type-{Q} {ABS} equations”, Int. Math. Res. Not., 2013 (2013), 1451–1481, arXiv: 1108.4502 | DOI | MR

[7] Baxter R. J., “The {Y}ang–{B}axter equations and the {Z}amolodchikov model”, Phys. D, 18 (1986), 321–347 | DOI | MR | Zbl

[8] Bazhanov V. V., Mangazeev V. V., Sergeev S. M., “Quantum geometry of three-dimensional lattices”, J. Stat. Mech. Theory Exp., 2008 (2008), P07004, 27 pp., arXiv: 0801.0129 | DOI | MR

[9] Behrend R. E., Pearce P. A., O'Brien D. L., “Interaction-round-a-face models with fixed boundary conditions: the {ABF} fusion hierarchy”, J. Statist. Phys., 84 (1996), 1–48, arXiv: hep-th/9507118 | DOI | MR | Zbl

[10] Bellon M. P., Viallet C.-M., “Algebraic entropy”, Comm. Math. Phys., 204 (1999), 425–437, arXiv: chao-dyn/9805006 | DOI | MR | Zbl

[11] Bobenko A. I., Suris Yu. B., “Integrable systems on quad-graphs”, Int. Math. Res. Not., 2002 (2002), 573–611, arXiv: nlin.SI/0110004 | DOI | MR | Zbl

[12] Bobenko A. I., Suris Yu. B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathematics, 98, American Mathematical Society, Providence, RI, 2008, arXiv: math.DG/0504358 | DOI | MR | Zbl

[13] Boll R., “Classification of 3{D} consistent quad-equations”, J. Nonlinear Math. Phys., 18 (2011), 337–365, arXiv: 1009.4007 | DOI | MR | Zbl

[14] Caudrelier V., Crampé N., Zhang Q. C., “Set-theoretical reflection equation: classification of reflection maps”, J. Phys. A: Math. Theor., 46 (2013), 095203, 12 pp., arXiv: 1210.5107 | DOI | MR | Zbl

[15] Caudrelier V., Zhang Q. C., Yang–Baxter and reflection maps from vector solitons with a boundary, arXiv: 1205.1133

[16] Cherednik I. V., “Factorizing particles on a half-line and root systems”, Theoret. and Math. Phys., 61 (1984), 977–983 | DOI | MR | Zbl

[17] Drinfeld V. G., “On some unsolved problems in quantum group theory”, Quantum Groups ({L}eningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8 | DOI | MR

[18] Fan H., Hou B. Y., Shi K. J., “General solution of reflection equation for eight-vertex {SOS} model”, J. Phys. A: Math. Gen., 28 (1995), 4743–4749 | DOI | MR | Zbl

[19] Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the {P}ainlevé property?, Phys. Rev. Lett., 67 (1991), 1825–1828 | DOI | MR | Zbl

[20] Habibullin I. T., Kazakova T. G., “Boundary conditions for integrable discrete chains”, J. Phys. A: Math. Gen., 34 (2001), 10369–10376 | DOI | MR | Zbl

[21] Hietarinta J., “Searching for {CAC}-maps”, J. Nonlinear Math. Phys., 12:2 (2005), 223–230 | DOI | MR | Zbl

[22] Levi D., Petrera M., Scimiterna C., “The lattice {S}chwarzian {K}d{V} equation and its symmetries”, J. Phys. A: Math. Theor., 40 (2007), 12753–12761, arXiv: math-ph/0701044 | DOI | MR | Zbl

[23] Levi D., Winternitz P., “Continuous symmetries of difference equations”, J. Phys. A: Math. Gen., 39 (2006), R1–R63, arXiv: nlin.SI/0502004 | DOI | MR | Zbl

[24] Mercat C., Holomorphie discrète et modèle d'Ising, Ph.D. Thesis, Université Louis Pasteur, Strasbourg, France, 1998 http://tel.archives-ouvertes.fr/tel-00001851/ | Zbl

[25] Mercat C., “Discrete {R}iemann surfaces and the {I}sing model”, Comm. Math. Phys., 218 (2001), 177–216 | DOI | MR | Zbl

[26] Nijhoff F.W., “Lax pair for the {A}dler (lattice {K}richever–{N}ovikov) system”, Phys. Lett. A, 297 (2002), 49–58, arXiv: nlin.SI/0110027 | DOI | MR | Zbl

[27] Papageorgiou V. G., Suris Yu. B., Tongas A. G., Veselov A. P., “On quadrirational {Y}ang–{B}axter maps”, SIGMA, 6 (2010), 033, 9 pp., arXiv: 0911.2895 | DOI | MR | Zbl

[28] Papageorgiou V. G., Tongas A. G., Veselov A. P., “Yang–{B}axter maps and symmetries of integrable equations on quad-graphs”, J. Math. Phys., 47 (2006), 083502, 16 pp., arXiv: math.QA/0605206 | DOI | MR | Zbl

[29] Rasin O. G., Hydon P. E., “Symmetries of integrable difference equations on the quad-graph”, Stud. Appl. Math., 119 (2007), 253–269 | DOI | MR

[30] Sklyanin E. K., “Boundary conditions for integrable quantum systems”, J. Phys. A: Math. Gen., 21 (1988), 2375–2389 | DOI | MR | Zbl