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

TL;DR

This paper provides complete proofs of theorems ensuring the consistency of discrete nonlinear equations defined by vanishing $3 imes 3$ determinants on cubic lattices, establishing conditions for initial data and demonstrating their validity.

Contribution

It offers rigorous proofs of consistency theorems for $3 imes 3$ determinant-based equations on cubic lattices, including explicit initial data conditions.

Findings

Proved theorems on consistency around a cube for these equations

Formulated explicit conditions of general position for initial data

Validated consistency for arbitrary initial data satisfying these conditions

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.