Weak Lax pairs for lattice equations

TL;DR

This paper explores the relationships between integrability concepts in 2D lattice equations, revealing that 3D consistency, Lax pairs, and Bäcklund transformations are not always equivalent, and introduces new lattice models and variants of the Yang-Baxter equation.

Contribution

It introduces weak Lax pairs for lattice equations and analyzes their connection with other integrability concepts, expanding the understanding of lattice integrability.

Findings

3D consistency, Lax pairs, and Bäcklund transformations are not strictly equivalent.

Introduces black and white lattice models and variants of the functional Yang-Baxter equation.

Provides insights into the nuanced relationships among integrability structures.

Abstract

We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and B\"acklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional Yang-Baxter equation.