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


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.

Previous article   Next article   Contents of Volume 7 (2011)