Integrable lattice equations with vertex and bond variables

TL;DR

This paper introduces new integrable lattice models with coupled vertex and bond variables, demonstrating their integrability through algebraic entropy and multidimensional consistency, and exploring models with both independent and coupled dynamics.

Contribution

The paper presents novel integrable lattice equations with coupled vertex and bond variables, including models with independent and fully coupled dynamics, analyzed via algebraic entropy and multidimensional consistency.

Findings

Models exhibit integrability confirmed by algebraic entropy.

Models demonstrate multidimensional consistency.

Vertex and bond variables can be either independent or fully coupled.

Abstract

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistency