Symmetries and integrability of discrete equations defined on a black-white lattice

TL;DR

This paper investigates the symmetries and integrability properties of deformed H equations on black-white lattices, revealing new discrete Toda equations, Bäcklund transformations, Lax pairs, and infinite symmetry hierarchies.

Contribution

It introduces two distinct three-leg forms for each equation, constructs associated discrete Toda equations, and analyzes their multidimensional consistency and symmetries, advancing understanding of discrete integrable systems.

Findings

Two different three-leg forms for each equation.

Construction of discrete Toda type equations.

Identification of Bäcklund transformations and Lax pairs.

Abstract

We study the deformations of the H equations, presented recently by Adler, Bobenko and Suris, which are naturally defined on a black-white lattice. For each one of these equations, two different three-leg forms are constructed, leading to two different discrete Toda type equations. Their multidimensional consistency leads to B{\"a}cklund transformations relating different members of this class, as well as to Lax pairs. Their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries.