From auto-B\"acklund transformations to auto-B\"acklund transformations, and torqued ABS equations

TL;DR

This paper introduces a method to generate new auto-Bäcklund transformations from existing ones, leading to torqued ABS equations and novel 3D consistent cubes, expanding the understanding of integrable discrete systems.

Contribution

The authors develop a systematic approach to derive new auto-Bäcklund transformations and apply it to ABS equations, revealing torqued versions and previously unknown 3D consistent cubes.

Findings

Derived torqued ABS equations from auto-BT transformations.

Discovered new 3D consistent cubes not previously identified.

Connected auto-BT transformations to multi-quadratic and multi-quartic equations.

Abstract

We provide a method which takes an auto-B\"acklund transformation (auto-BT) and produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions of ABS equations and explains the origin of each auto-BT listed in [J. Atkinson, J. Phys. A: Math. Theor. 41 (2008) 135202]. The method is also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes which have not been found in [R. Boll, J. Nonl. Math. Phys. 18 (2011) 337--365], and to a multi-quadratic ABS* equation giving rise to a multi-quartic equation.