Integrability of auto-B\"acklund transformations,and solutions of a torqued ABS equation

TL;DR

This paper explores the integrability of an auto-B"acklund transformation related to a torqued version of the H2 equation, deriving solutions and revealing its semi-autonomous lattice structure.

Contribution

It introduces a new torqued version of the H2 equation, constructs its auto-B"acklund transformation and Lax pair, and derives explicit solutions demonstrating semi-autonomy.

Findings

Derived a seed and one-soliton solution for H2^a.

Showed H2^a is a semi-autonomous lattice equation.

Established the auto-B"acklund transformation admits auto-B"acklund transformations.

Abstract

An auto-B\"acklund transformation for the quad equation $\mathrm{Q1}_1$ is considered as a discrete equation, called $\mathrm{H2}^a$, which is a so called torqued version of $\mathrm{H2}$. The equations $\mathrm{H2}^a$ and $\mathrm{Q1}_1$ compose a consistent cube, from which a auto-B\"acklund transformation and a Lax pair for $\mathrm{H2}^a$ are obtained. More generally it is shown that auto-B\"acklund transformations admit auto-B\"acklund transformations. Using the auto-B\"acklund transformation for $\mathrm{H2}^a$ we derive a seed solution and a one-soliton solution. From this solution it is seen that $\mathrm{H2}^a$ is a semi-autonomous lattice equation, as the spacing parameter $q$ depends on $m$ but it disappears from the plain wave factor.