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SIGMA, 2018, Volume 14, 059, 14 pages (Mi sigma1358)  

This article is cited in 1 scientific paper (total in 1 paper)

Dressing the Dressing Chain

Charalampos A. Evripidoua, Peter H. van der Kampa, Cheng Zhangb

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

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 $n$.

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

Funding Agency Grant Number
Australian Research Council
National Natural Science Foundation of China 11601312
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).


DOI: https://doi.org/10.3842/SIGMA.2018.059

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Bibliographic databases:

Document Type: Article
MSC: 35Q53; 37K05; 39A14
Received: April 18, 2018; in final form June 4, 2018; Published online June 15, 2018
Language: English

Citation: Charalampos A. Evripidou, Peter H. van der Kamp, Cheng Zhang, “Dressing the Dressing Chain”, SIGMA, 14 (2018), 059, 14 pp.

Citation in format AMSBIB
\Bibitem{EvrVanZha18}
\by Charalampos~A.~Evripidou, Peter~H.~van der Kamp, Cheng~Zhang
\paper Dressing the Dressing Chain
\jour SIGMA
\yr 2018
\vol 14
\papernumber 059
\totalpages 14
\mathnet{http://mi.mathnet.ru/sigma1358}
\crossref{https://doi.org/10.3842/SIGMA.2018.059}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85050351343}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. G. S. Mauleshova, “The dressing chain and one-point commuting difference operators of rank 1”, Siberian Math. J., 59:5 (2018), 901–908  mathnet  crossref  crossref  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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