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

SIGMA 10 (2014), 014, 24 pages      arXiv:1307.4023      https://doi.org/10.3842/SIGMA.2014.014

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

Abstract
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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References

  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 http://tel.archives-ouvertes.fr/tel-00001851/.
  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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