A noncommutative discrete potential KdV lift

TL;DR

This paper introduces a noncommutative extension of a Yang-Baxter map related to the discrete potential KdV equation, demonstrating its integrability, Lax representation, and connection to lattice systems.

Contribution

It constructs a Grassmann extension of a Yang-Baxter map that lifts the dpKdV equation and satisfies the Yang-Baxter equation, with a Lax matrix and integrability properties.

Findings

Constructed a Grassmann extension satisfying Yang-Baxter equation

Derived a lattice system with Lax representation

Discussed integrability of commutative analogues

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.