### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 059, 14 pages      arXiv:1804.02564      https://doi.org/10.3842/SIGMA.2018.059

### Dressing the Dressing Chain

Charalampos A. Evripidou a, Peter H. van der Kamp a and Cheng Zhang b
a) Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia
b) Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China

Received April 18, 2018, in final form June 04, 2018; Published online June 15, 2018

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-Bäcklund 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-Bäcklund 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 $n$.

Key words: discrete dressing chain; lattice KdV; Darboux transformations; Liouville integrability.

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