Classification of 3D consistent quad-equations. (English) Zbl 1252.37068
J. Nonlinear Math. Phys. 18, No. 3, 337-365 (2011); corrigendum ibid. 19, No. 4, Paper No. 1292001 (2012).
Summary: We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice \(\mathbb Z^{3}\) is possible by reflecting the cubes. For every quad-equation we will give at least one system included leading to a Bäcklund transformation and a zero-curvature representation which means that they are integrable.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
integrability; quad-graph; multidimensional consistency; zero curvature representation; Bäcklund transformation; Möbius transformationCitations:
Zbl 1262.37033References:
| [1] | Nijhoff, F. W.; Walker, A. J., The discrete and continous Painlevé VI hierachy and the Garnier systems, Glasg, Math. J, 43A, 109-123, 2001 · Zbl 0990.39015 |
| [2] | Bobenko, A. I.; Suris, Y. B., Integrable systems on quad-graphs, Int. Math. Res. Not, 11, 573-611, 2002 · Zbl 1004.37053 |
| [3] | Nijhoff, F. W., Lax pair for Adler (lattice Krichever-Novikov) system, Phys. Lett, 297, 49-58, 2002 · Zbl 0994.35105 |
| [4] | Adler, V. E.; Bobenko, A. I.; Suris, Y. B., Classification of integrable equations on quadgraphs. The consistency approach, Comm, Math. Phys, 233, 513-543, 2003 · Zbl 1075.37022 |
| [5] | Atkinson, J., Bäcklund transformations for integrable lattice equations, 2008 · Zbl 1148.82006 |
| [6] | Boll, R.; Suris, Y. B., Non-symmetric discrete Toda systems from quad-graphs, Appl, Anal, 89, 4, 547-569, 2010 · Zbl 1189.35270 |
| [7] | Adler, V. E.; Bobenko, A. I.; Suris, Y. B., Discrete nonlinear hyperbolic equations. Classification of integrable cases, Funct. Anal, Appl, 43, 3-17, 2009 · Zbl 1271.37048 |
| [8] | Hietarinta, J., A new two-dimensional lattice model that is “consistent around a cube”, J. Phys. A: Math. Theor, 37, 67-73, 2004 · Zbl 1044.81101 |
| [9] | Hydon, P. E.; Viallet, C-M, Asymmetric integrable quad-graph equations, Appl, Anal, 89, 4, 493-506, 2010 · Zbl 1188.37059 |
| [10] | Levi, D.; Yamilov, R. I., On a linear inegrable difference equation on the square, Ufa, Math. J, 1, 2, 101-105, 2009 · Zbl 1240.39020 |
| [11] | Xenitidis, P. D.; Papageorgiou, V. G., Symmetries and integrability of discrete equations defined on a black-white lattice, 2009 |
| [12] | Bobenko, A. I.; Suris, Y. B., Discrete Differential Geometry. Integrable Structure, Graduate Studies in Mathematics, Vol, 2008 · Zbl 1158.53001 |
| [13] | Iatrou, A.; Roberts, J. A G., Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 14, 459-489, 2002 · Zbl 1067.37072 |
| [14] | Atkinson, J., J. Phys. A:, 2009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
