Dressing the Dressing Chain

TL;DR

This paper explores the integrability and transformations of the dressing chain, linking it to the lattice KdV equation and auto-B"acklund transformations, with a focus on periodic reductions and explicit formulas.

Contribution

It demonstrates how Darboux transformations generate the lattice KdV from the dressing chain and establishes integrability for odd periodic reductions.

Findings

Darboux transformations relate the dressing chain to lattice KdV.

The lattice KdV acts as an auto-B"acklund transformation.

Complete integrability is proven for odd n in periodic reductions.

Abstract

The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-B\"acklund transformation for the modified KdV equation. We show that by applying Darboux transformations to the spectral problem of the dressing chain one obtains the lattice KdV equation as the dressing chain of the dressing chain and, that the lattice KdV equation also arises as an auto-B\"acklund transformation for a modified dressing chain. In analogy to the results obtained for the dressing chain (Veselov and Shabat proved complete integrability for odd dimensional periodic reductions), we study the $(0,n)$-periodic reduction of the lattice KdV equation, which is a two-valued correspondence. We provide explicit formulas for its branches and establish complete integrability for odd $n$.