On Initial Data in the Problem of Consistency on Cubic Lattices for 3×3 Determinants

[摘要] 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.

[关键词] consistency principle;square and cubic lattices;integrable discrete equation;initial data;determinant;bent elementary square;consistency “around a cube”