Continuous symmetric reductions of the Adler-Bobenko-Suris equations

TL;DR

This paper introduces a method for finding continuous symmetric solutions to the Adler-Bobenko-Suris discrete integrable equations, linking them to PDE systems, and deriving explicit solutions involving Painleve transcendents.

Contribution

It develops a novel approach to generate continuous symmetric solutions for ABS equations, connecting them to integrable PDEs and providing explicit solutions with Painleve transcendents.

Findings

Solutions are determined by an integrable PDE system.

Connection established with Nijhoff-Hone-Joshi generating PDEs.

Explicit solutions involve Painleve transcendents.

Abstract

Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi "generating partial differential equations" is established and an auto-Backlund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1$_{\delta=0}$ members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents.