Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

S. Konstantinou-Rizos, A. A. Kutuzova

Abstract

We construct noncommutative maps related to the Boussinesq and Nonlinear Schrödinger (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.

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

Abstract

We construct noncommutative maps related to the Boussinesq and Nonlinear Schrödinger (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.
Paper Structure (11 sections, 8 theorems, 83 equations, 1 figure)

This paper contains 11 sections, 8 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Set-theoretical Yang--Baxter equation and Lax representations
  5. Zamolodchikov tetrahedron and local Yang--Baxter equation
  6. Boussinesq type 2-simplex maps
  7. Commutative Boussinesq type Yang--Baxter map
  8. Nonommutative Boussinesq type Yang--Baxter maps
  9. Squeeze down to the noncommutative Boussinesq lattice equation
  10. NLS type 3-simplex map
  11. Conclusions

Key Result

Theorem 2.1

(Kouloukas--Papageorgiou Kouloukas) Let $Y_{a,b}$ be a map with Lax representation eq-Lax. If the following matrix trifactorisation problem implies $u=x$, $v=y$, $w=z$, then $Y_{a,b}$ satisfies the parametric Yang--Baxter equation ParamYB.

Figures (1)

  • Figure 1: Quad graph equation.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 4 more