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

Abstract

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

References in corpus (4)

• Backlund transformations for integrable lattice equations
• Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation
• Multi-Component Extension of CAC Systems
• From auto-Bäcklund transformations to auto-Bäcklund transformations, and torqued ABS equations