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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”.
Received: January 23, 2011; in final form July 17, 2011; Published online July 26, 2011
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.
Linking options:
https://www.mathnet.ru/eng/sigma633 https://www.mathnet.ru/eng/sigma/v7/p75
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