SIGMA 7 (2011), 075, 19 pages      arXiv:1101.4355      https://doi.org/10.3842/SIGMA.2011.075
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

On Initial Data in the Problem of Consistency on Cubic Lattices for 3×3 Determinants

Oleg I. Mokhov ^{a, b
a)} Centre for Nonlinear Studies, L.D.Landau Institute for Theoretical Physics, Russian Academy of Sciences, 2 Kosygina Str., Moscow, Russia
^b) Department of Geometry and Topology, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow, Russia

Received January 23, 2011, in final form July 17, 2011; Published online July 26, 2011

Abstract
The paper is devoted to complete proofs of theorems on consistency on cubic lattices for 3×3 determinants. The discrete nonlinear equations on Z² defined by the condition that the determinants of all 3×3 matrices of values of the scalar field at the points of the lattice Z² that form elementary 3×3 squares vanish are considered; some explicit concrete conditions of general position on initial data are formulated; and for arbitrary initial data satisfying these concrete conditions of general position, theorems on consistency on cubic lattices (a consistency ''around a cube'') for the considered discrete nonlinear equations on Z² defined by 3×3 determinants are proved.

Key words: consistency principle; square and cubic lattices; integrable discrete equation; initial data; determinant; bent elementary square; consistency ''around a cube''.

pdf (345 kb)   tex (18 kb)

References

1. Mokhov O.I., On consistency of determinants on cubic lattices, Uspekhi Mat. Nauk 63 (2008), no. 6, 169-170 (English transl.: Russian Math. Surveys 63 (2008), no. 6, 1146-1148), arXiv:0809.2032.
2. Mokhov O.I., Consistency on cubic lattices for determinants of arbitrary orders, in Geometry, Topology and Mathematical Physics, Tr. Mat. Inst. Steklova 266 (2009), 202-217 (English transl.: Proc. Steklov Inst. Math. 266 (2009), 195-209), arXiv:0910.2044.
3. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43A (2001), 109-123, nlin.SI/0001054.
4. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
5. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), no. 11, 573-611, nlin.SI/0110004.
6. 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.
7. 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.
8. Bobenko A.I., Suris Yu.B., On organizing principles of discrete differential geometry. Geometry of spheres, Uspekhi Mat. Nauk 62 (2007), no. 1, 3-50 (English transl.: Russian Math. Surveys 62 (2007), no. 1, 1-43), math.DG/0608291.
9. Veselov A.P., Integrable maps, Uspekhi Mat. Nauk 46 (1991), no. 5, 3-45 (English transl.: Russian Math. Surveys 46 (1991), no. 5, 1-51).
10. Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations: classification of integrable cases, Funct. Anal. Appl. 43 (2009), 3-17, arXiv:0705.1663.
11. Tsarev S.P., Wolf Th., Classification of three-dimensional integrable scalar discrete equations, Lett. Math. Phys 84 (2008), 31-39, arXiv:0706.2464.
12. Hietarinta J., A new two-dimensional lattice model that is `consistent around a cube', J. Phys. A: Math. Gen. 37 (2004), L67-L73, nlin.SI/0311034.
13. Adler V.E., Veselov A.P., Cauchy problem for integrable discrete equations on quad-graphs, Acta Appl. Math. 84 (2004), 237-262, math-ph/0211054.
14. Adler V.E., Suris Yu.B., Q₄: integrable master equation related to an elliptic curve, Int. Math. Res. Not. 2004 (2004), no. 47, 2523-2553, nlin.SI/0309030.