Group Analysis of Differential Equations and Integrable Systems − 2018


Charalambos Evripidou (La Trobe University, Bundoora, Victoria, Australia)

Dressing the dressing chain

Abstract:
The dressing chain (DC) arises in the application of Darboux transformations to the Schödinger equation, which is the spectral problem for the Korteweg-de Vries (KdV) equation. It also arises as an auto-Bäcklund transformation of the modified KdV equation (mKdV) via the celebrated Miura transformation between KdV and mKdV. Odd dimensional periodic reductions of the DC were shown to be Liouville integrable by Veselov and Shabat. In this talk I will demonstrate what happens if we replace the KdV by the DC in the above text: The lattice KdV equation (lKdV) arises in the application of Darboux transformations to the discrete Schödinger equation, which is the spectral problem for the DC. It also arises as an auto-Bäcklund transformation of a modified DC (mDC) via a Bäcklund transformation between DC and mDC. Odd dimensional periodic reductions of the lKdV equation are shown to be Liouville integrable.