Grassmann extensions of Yang-Baxter maps

Georgi G. Grahovski, Sotiris Konstantinou-Rizos, Alexander V. Mikhailov

Published 2015-10-23Version 1

In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassman extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrodinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. 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. We consider their vector generalisations.