Grassmann extensions of Yang-Baxter maps

TL;DR

This paper introduces explicit Yang-Baxter maps with Darboux-Lax representations on Grassmann-extended algebraic varieties, deriving new endomorphisms and maps related to Grassmann-extended NLS and DNLS equations.

Contribution

It presents novel Grassmann-extended Yang-Baxter maps and their Darboux matrix derivation, expanding the understanding of noncommutative integrable systems.

Findings

Derived Darboux matrix for Grassmann-extended DNLS

Constructed 10D Yang-Baxter maps on Grassmann varieties

Restricted maps to 8D invariant leaves related to NLS and DNLS

Abstract

In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann 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.