Geodesic
On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3\times3$ determinants. The discrete nonlinear equations on $\mathbb{Z}^2$ defined by the condition that the determinants of all $3\times3$ matrices of values of the scalar field at the points of the lattice $\mathbb{Z}^2$ that form elementary $3\times3$ 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 $\mathbb{Z}^2$ defined by $3\times3$ determinants are proved.
Keywords: consistency principle; square and cubic lattices; integrable discrete equation; initial data; determinant; bent elementary square; consistency “around a cube”. 
@article{SIGMA_2011_7_a74,
     author = {Oleg I. Mokhov},
     title = {On {Initial} {Data} in the {Problem} of {Consistency} on {Cubic} {Lattices} for $3\times3$ {Determinants}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2011},
     volume = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a74/}
}
TY  - JOUR
AU  - Oleg I. Mokhov
TI  - On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2011
VL  - 7
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a74/
LA  - en
ID  - SIGMA_2011_7_a74
ER  - 
%0 Journal Article
%A Oleg I. Mokhov
%T On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants
%J Symmetry, integrability and geometry: methods and applications
%D 2011
%V 7
%U http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a74/
%G en
%F SIGMA_2011_7_a74
Oleg I. Mokhov. On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a74/

[1] Russian Math. Surveys, 63:6 (2008), 1146–1148, arXiv: 0809.2032 | DOI | MR | Zbl

[2] Proc. Steklov Inst. Math., 266, 2009, 195–209, arXiv: 0910.2044 | DOI | MR | Zbl

[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, arXiv: nlin.SI/0001054 | DOI | MR | Zbl

[4] Nijhoff F. W., “Lax pair for the Adler (lattice Krichever–Novikov) system”, Phys. Lett. A, 297 (2002), 49–58, arXiv: nlin.SI/0110027 | DOI | MR | Zbl

[5] Bobenko A. I., Suris Yu. B., “Integrable systems on quad-graphs”, Int. Math. Res. Not., 2002:11 (2002), 573–611, arXiv: nlin.SI/0110004 | DOI | MR | Zbl

[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, arXiv: nlin.SI/0202024 | DOI | MR | Zbl

[7] Bobenko A. I., Suris Yu. B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathematics, 98, American Mathematical Society, Providence, RI, 2008, arXiv: math.DG/0504358 | MR | Zbl

[8] Russian Math. Surveys, 62:1 (2007), 1–43, arXiv: math.DG/0608291 | DOI | MR | Zbl

[9] Russian Math. Surveys, 46:5 (1991), 1–51 | DOI | MR | Zbl

[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 | DOI | MR

[11] Tsarev S. P., Wolf Th., “Classification of three-dimensional integrable scalar discrete equations”, Lett. Math. Phys, 84 (2008), 31–39, arXiv: 0706.2464 | DOI | MR | Zbl

[12] Hietarinta J., “A new two-dimensional lattice model that is ‘consistent around a cube’”, J. Phys. A: Math. Gen., 37 (2004), L67–L73, arXiv: nlin.SI/0311034 | DOI | MR | Zbl

[13] Adler V. E., Veselov A. P., “Cauchy problem for integrable discrete equations on quad-graphs”, Acta Appl. Math., 84 (2004), 237–262, arXiv: math-ph/0211054 | DOI | MR | Zbl

[14] Adler V. E., Suris Yu. B., “${\rm Q}_4$: integrable master equation related to an elliptic curve”, Int. Math. Res. Not., 2004:47 (2004), 2523–2553, arXiv: nlin.SI/0309030 | DOI | MR | Zbl