Dressing the Dressing Chain
journal contributionposted on 2022-03-18, 02:58 authored by Charalambos EvripidouCharalambos Evripidou, Pieter Van Der KampPieter Van Der Kamp, Cheng Zhang
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.