Difference systems in bond and face variables and non-potential versions of discrete integrable systems

TL;DR

This paper explores difference systems in bond and face variables for discrete integrable equations, revealing non-potential forms, higher-point configurations, and multiquadratic relations on 2D and 3D lattices.

Contribution

It introduces a unified approach to rewrite discrete equations as vector systems, linking equations via difference substitutions and uncovering new multi-point and multiquadratic integrable relations.

Findings

Linking discrete equations through difference substitutions.

Derivation of higher-point compatible equations on lattices.

Presentation of integrable multiquadratic quad relations.

Abstract

Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the difference systems in face variables can be regarded as vector versions of the original equations. As a result, we link some of the discrete equations by difference substitutions and reveal the non-potential versions of some consistent-around-the-cube equations. We obtain higher-point configurations, including pairs of compatible six~points equations on the ${\mathbb Z}^2$ lattice together with associated seven points equations. Also we obtain a variety of compatible ten point equations together with associated ten and twelve point equations on the ${\mathbb Z}^3$ lattice. Finally, we present integrable multiquadratic quad relations.