
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 (SIDE9)”
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 Z^{2} defined by the condition that
the determinants of all 3×3 matrices of values of the
scalar field at the points of the lattice Z^{2} 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 Z^{2} 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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