Bianchi permutability for the anti-self-dual Yang-Mills equations

TL;DR

This paper develops a Bäcklund transformation for the anti-self-dual Yang-Mills equations, enabling the generation of new solutions and connecting to integrable systems through Bianchi permutability and reductions.

Contribution

It introduces a general Bäcklund transformation for ASDYM equations with a Darboux matrix, and explores its applications to reductions and discrete integrable systems.

Findings

Derived a Bäcklund transformation with Bianchi permutability for ASDYM

Obtained new solutions via symmetry reductions of ASDYM

Connected ASDYM reductions to discrete integrable systems

Abstract

The anti-self-dual Yang-Mills equations are known to have reductions to many integrable differential equations. A general B\"acklund transformation (BT) for the ASDYM equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.