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

Abstract

Abstract An auto-Bäcklund transformation for the quad equation Q11 is considered as a discrete equation, called H2 a , which is a so called torqued version of H2. The equations H2 a and Q11 compose a consistent cube, from which an 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 plane wave factor.