Grassmann extensions of Yang-Baxter maps

2015/10/31 by Georgi G. Grahovski, Sotiris Konstantinou-Rizos, Alexander V. Mikhailov · 2 citations
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.MP

paper · pdf · doi:10.1088/1751-8113/49/14/145202

published as J. Phys. A: Math. Theor. 49 (2016) 145202 · 18 pages, LaTeX

arxiv created 2016/02/23 · arxiv updated 2016/02/25

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.

Cited by

• From NLS type matrix refactorisation problems to set-theoretical solutions of the 2- and 3-simplex equations
• Multicomponent pentagon maps

Related

• Entwining Yang-Baxter maps over Grassmann algebras
• Darboux transformations, finite reduction groups and related Yang-Baxter maps
• Integrable extensions of Adler's map via Grassmann algebras
• Yang--Baxter maps of KdV, NLS and DNLS type on division rings
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps