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ЖУРНАЛЫ // Symmetry, Integrability and Geometry: Methods and Applications // Архив

SIGMA, 2011, том 7, 075, 19 стр. (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

Аннотация: 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.

Ключевые слова: consistency principle; square and cubic lattices; integrable discrete equation; initial data; determinant; bent elementary square; consistency “around a cube”.

MSC: 39A05; 52C07; 15A15; 37K10; 11H06

Поступила: 23 января 2011 г.; в окончательном варианте 17 июля 2011 г.; опубликована 26 июля 2011 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2011.075



Реферативные базы данных:
ArXiv: 1101.4355


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