Xueli Wei

Integrability of auto-Bäcklund transformations,and solutions of a torqued ABS equation

An auto-Bäcklund 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-Bäcklund transformation and a Lax pair for $\mathrm{H2}^a$ are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund 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.