Dressing the Dressing Chain

Symmetry Integrability and Geometry-Methods and Applications

article

Charalampos A. Evripidou¹  Peter H. van der Kamp¹  Cheng Zhang² 

[1] Department of Mathematics and Statistics, La Trobe University;Department of Mathematics, Shanghai University

关键词: discrete dressing chain;    lattice KdV;    Darboux transformations;    Liouville integrability;   

DOI  :  10.3842/SIGMA.2018.059

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $n$.

【 授权许可】

Unknown