Solutions of Adler's lattice equation associated with 2-cycles of the Backlund transformation

TL;DR

This paper explores special solutions of Adler's lattice equation that are invariant under a composition of two Bäcklund transformations, revealing their connection to commuting two-variable mappings and providing explicit solutions.

Contribution

It introduces a novel link between 2-cycle solutions of Adler's lattice equation and commuting rank-2 mappings, offering explicit solutions for these special cases.

Findings

2-cycle solutions are associated with commuting two-variable mappings

Explicit solutions of the mappings are constructed

Provides solutions of Adler's equation that are 2-cycles of the BT

Abstract

The BT of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Backlund parameters) as a 2-cycle of the BT. In this article we will show that such solutions are associated with a commuting one-parameter family of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit solution of the mappings within this family and hence give the solutions of Adler's equation that are 2-cycles of the BT.