On Initial Data in the Problem of Consistency on Cubic Lattices for $3 \times 3$ Determinants

Abstract

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3 \times 3$ determinants. The discrete nonlinear equations on $\mathbb{Z}^2$ defined by the condition that the determinants of all $3 \times 3$ matrices of values of the scalar field at the points of the lattice $\mathbb{Z}^2$ that form elementary $3 \times 3$ 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 $\mathbb{Z}^2$ defined by $3 \times 3$ determinants are proved.