On decomposition of the ABS lattice equations and related B\"acklund transformations

TL;DR

This paper revisits the decomposition of ABS lattice equations, constructing Bäcklund transformations and related structures, leading to new solutions, equations, and Lax pairs in integrable systems.

Contribution

It provides a detailed analysis of ABS lattice equations' decomposition, enabling the construction of Bäcklund transformations and new integrable structures.

Findings

New Bäcklund transformations derived from decompositions

Construction of new solutions and lattice equations

Development of weak Lax pairs for integrability

Abstract

The Adler-Bobenko-Suris (ABS) list contains all scalar quadrilateral equations which are consistent around the cube. Each equation in the ABS list admits a beautiful decomposition. In this paper, we first revisit these decomposition formulas, by which we construct B\"acklund transformations (BTs) and consistent triplets. Some BTs are used to construct new solutions, lattice equations and weak Lax pairs.