Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 014, 24 pages      arXiv:1307.4023

Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency

Vincent Caudrelier a, Nicolas Crampé b and Qi Cheng Zhang a
a) Department of Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK
b) CNRS, Laboratoire Charles Coulomb, UMR 5221, Place Eugène Bataillon - CC070, F-34095 Montpellier, France

Received July 19, 2013, in final form February 05, 2014; Published online February 12, 2014

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.

Key words: discrete integrable systems; quad-graph equations; 3D-consistency; Bäcklund transformations; zero curvature representation; Toda-type systems; set-theoretical reflection equation.

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  1. Adler V.E., Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453-10460.
  2. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  3. Adler V.E., Bobenko A.I., Suris Yu.B., Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings, Comm. Anal. Geom. 12 (2004), 967-1007, math.QA/0307009.
  4. Ahn C., Koo W.M., Boundary Yang-Baxter in the RSOS/SOS representation, in Statistical Models, Yang-Baxter Equation and Related Topics, and Symmetry, Statistical Mechanical Models and Applications (Tianjin, 1995), World Sci. Publ., River Edge, NJ, 1996, 3-12, hep-th/9508080.
  5. Atkinson J., Hietarinta J., Nijhoff F., Seed and soliton solutions for Adler's lattice equation, J. Phys. A: Math. Theor. 40 (2007), F1-F8, nlin.SI/0609044.
  6. Atkinson J., Joshi N., Singular-boundary reductions of type-Q ABS equations, Int. Math. Res. Not. 2013 (2013), 1451-1481, arXiv:1108.4502.
  7. Baxter R.J., The Yang-Baxter equations and the Zamolodchikov model, Phys. D 18 (1986), 321-347.
  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 pages, arXiv:0801.0129.
  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, hep-th/9507118.
  10. Bellon M.P., Viallet C.-M., Algebraic entropy, Comm. Math. Phys. 204 (1999), 425-437, chao-dyn/9805006.
  11. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, nlin.SI/0110004.
  12. Bobenko A.I., Suris Yu.B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathematics, Vol. 98, American Mathematical Society, Providence, RI, 2008, math.DG/0504358.
  13. Boll R., Classification of 3D consistent quad-equations, J. Nonlinear Math. Phys. 18 (2011), 337-365, arXiv:1009.4007.
  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 pages, arXiv:1210.5107.
  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.
  17. Drinfeld V.G., On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 1-8.
  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.
  19. Grammaticos B., Ramani A., Papageorgiou V., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), 1825-1828.
  20. Habibullin I.T., Kazakova T.G., Boundary conditions for integrable discrete chains, J. Phys. A: Math. Gen. 34 (2001), 10369-10376.
  21. Hietarinta J., Searching for CAC-maps, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 223-230.
  22. Levi D., Petrera M., Scimiterna C., The lattice Schwarzian KdV equation and its symmetries, J. Phys. A: Math. Theor. 40 (2007), 12753-12761, math-ph/0701044.
  23. Levi D., Winternitz P., Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006), R1-R63, nlin.SI/0502004.
  24. Mercat C., Holomorphie discrète et modèle d'Ising, Ph.D. Thesis, Université Louis Pasteur, Strasbourg, France, 1998, available at
  25. Mercat C., Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), 177-216.
  26. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
  27. Papageorgiou V.G., Suris Yu.B., Tongas A.G., Veselov A.P., On quadrirational Yang-Baxter maps, SIGMA 6 (2010), 033, 9 pages, arXiv:0911.2895.
  28. Papageorgiou V.G., Tongas A.G., Veselov A.P., Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys. 47 (2006), 083502, 16 pages, math.QA/0605206.
  29. Rasin O.G., Hydon P.E., Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119 (2007), 253-269.
  30. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.

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