Dressing the Dressing Chain

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 $n$.

References in corpus (6)

• Discrete integrable systems and Poisson algebras from cluster maps
• The Generalized Symmetry Method for Discrete Equations
• Integrable and superintegrable systems associated with multi-sums of products
• Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems
• Integrable Discrete Nonautonomous Quad-equations as Bäcklund Auto-transformations for Known Volterra and Toda Type Semidiscrete Equations
• Discrete Crum's Theorems and Integrable Lattice Equations