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.
Funding
This work was supported by the Australian Research Council, by the China Strategy Implementation Grant Program of La Trobe University, by the NSFC (No. 11601312) and by the Shanghai Young Eastern Scholar program (2016-2019).
History
Publication Date
2018-06-15
Journal
Symmetry Integrability and Geometry: Methods and Applications (SIGMA)
Volume
14
Article Number
059
Pagination
14p.
Publisher
Department of Applied Research, Institute of Mathematics of National Academy of Sciences of Ukraine
ISSN
1815-0659
Rights Statement
“The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License.”