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 SIGMA, 2011, Volume 7, 075, 19 pp. (Mi sigma633)

On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants

Oleg I. Mokhovab

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

Abstract: 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”

DOI: https://doi.org/10.3842/SIGMA.2011.075

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Full text: http://emis.mi.ras.ru/journals/SIGMA/2011/075/
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Bibliographic databases:

ArXiv: 1101.4355
MSC: 39A05; 52C07; 15A15; 37K10; 11H06
Received: January 23, 2011; in final form July 17, 2011; Published online July 26, 2011
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Citation: Oleg I. Mokhov, “On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants”, SIGMA, 7 (2011), 075, 19 pp.

Citation in format AMSBIB
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\by Oleg I.~Mokhov
\paper On Initial Data in the Problem of Consistency on Cubic Lattices for $3\times3$ Determinants
\jour SIGMA
\yr 2011
\vol 7
\papernumber 075
\totalpages 19
\mathnet{http://mi.mathnet.ru/sigma633}
\crossref{https://doi.org/10.3842/SIGMA.2011.075}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2861201}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896063064}