Noncommutative Boussinesq and NLS type 2- and 3-simplex maps

TL;DR

This paper constructs noncommutative integrable maps related to Boussinesq and NLS equations, demonstrating their satisfaction of key algebraic equations and deriving lattice and tetrahedron map versions.

Contribution

It introduces noncommutative versions of Boussinesq and NLS maps, showing their relation to Yang-Baxter and Zamolodchikov tetrahedron equations.

Findings

Noncommutative Boussinesq map satisfies Yang-Baxter equation.

Noncommutative Boussinesq map reduces to a lattice equation.

Noncommutative NLS map is a Zamolodchikov tetrahedron map.

Abstract

We construct noncommutative maps related to the Boussinesq and Nonlinear Schr\"odinger (NLS) equations with their variables belonging to a noncommutative division ring. We show that the noncommutative Boussinesq type map satisfies the Yang--Baxter equation, and it can be squeezed down to a noncommutative version of the Boussinesq lattice equation. Moreover, we show that the noncommutative NLS type map is a Zamolodchikov tetrahedron map.