A noncommutative discrete potential KdV lift

Abstract

In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in [16] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a $3\times 3$ Lax matrix. Moreover, we show that it can be squeezed down to a system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogues of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability.

16 pages, 1 figure

References in corpus (5)

• Grassmann extensions of Yang-Baxter maps
• Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
• Yang-Baxter maps associated to elliptic curves
• Anticommutative extension of the Adler map
• Poisson structures for lifts and periodic reductions of integrable lattice equations

Cited by in corpus (7)

• On the $3D$ consistency of a Grassmann extended lattice Boussinesq system
• Entwining Yang-Baxter maps related to NLS type equations
• The Coxeter relations and KP map for non-commuting symbols
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Birational solutions to the set-theoretical 4-simplex equation