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


SIGMA 7 (2011), 075, 19 pages      arXiv:1101.4355      http://dx.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''.

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