Georgi G. Grahovski - University of Essex
Résumé : A review of integrable discretisations for a class of coupled nonlinear Schr¨odinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are presented. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting discrete systems will have Lax pairs provided by the set of two consistent Darboux transformations. Finally, Yang-Baxter maps for the Grassmann-extended NLS equation will be presentes. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. Their Liouville integrability will be briefly discussed. Based on a joint work with A. V. Mikhailov and S. G. Konstantinou-Rizos.
