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

Sotiris Konstantinou-Rizos

TL;DR

The work tackles the construction of set-theoretical $n$-simplex maps by varying the spectral parameter in Lax (Darboux) matrices, producing hierarchies of Yang–Baxter and Zamolodchikov maps tied to Adler and NLS-type equations. The authors derive explicit Adler-type YB maps $Y_1$ and $Y_2$, NLS-type maps $Y_3$ and $Y_4$, derivative NLS-type maps $Y_5$ and $Y_6$, and a parametric Zamolodchikov tetrahedron map $T_{a,b,c}$, many of which come with integrals of motion. They prove Liouville integrability for at least two maps ($Y_1$ and $Y_6$) and establish invariant structures for others, demonstrating the viability of the spectral-parameter variation method for generating new integrable maps. The results extend prior work on Darboux-refactorisation and local simplex equations, offering a framework for noncommutative extensions and higher-simplex or polygon generalisations with potential applications in integrable systems theory.

Abstract

We present a method for constructing hierarchies of solutions to $n$-simplex equations by variating the spectral parameter in their Lax representation. We use this method to derive new solutions to the set-theoretical 2- and 3-simplex equations which are related to the Adler map and Nonlinear Schrödinger (NLS) type equations. Moreover, we prove that some of the derived Yang--Baxter maps are completely integrable.

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

TL;DR

The work tackles the construction of set-theoretical -simplex maps by varying the spectral parameter in Lax (Darboux) matrices, producing hierarchies of Yang–Baxter and Zamolodchikov maps tied to Adler and NLS-type equations. The authors derive explicit Adler-type YB maps and , NLS-type maps and , derivative NLS-type maps and , and a parametric Zamolodchikov tetrahedron map , many of which come with integrals of motion. They prove Liouville integrability for at least two maps ( and ) and establish invariant structures for others, demonstrating the viability of the spectral-parameter variation method for generating new integrable maps. The results extend prior work on Darboux-refactorisation and local simplex equations, offering a framework for noncommutative extensions and higher-simplex or polygon generalisations with potential applications in integrable systems theory.

Abstract

We present a method for constructing hierarchies of solutions to -simplex equations by variating the spectral parameter in their Lax representation. We use this method to derive new solutions to the set-theoretical 2- and 3-simplex equations which are related to the Adler map and Nonlinear Schrödinger (NLS) type equations. Moreover, we prove that some of the derived Yang--Baxter maps are completely integrable.
Paper Structure (10 sections, 4 theorems, 57 equations)

This paper contains 10 sections, 4 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Yang--Baxter and tetrahedron maps
  3. Yang--Baxter and local 1-simplex equation
  4. Zamolodchikov tetrahedron and local 2-simplex equation
  5. Variation of the spectral parameter
  6. Darboux transformations and Yang--Baxter maps
  7. NLS type Yang--Baxter maps
  8. Derivative NLS type Yang--Baxter maps
  9. Darboux transformations and Zamolodchikov maps
  10. Discussion

Key Result

Theorem 4.2

Map $Y_3:\mathbb{C}^6\rightarrow \mathbb{C}^6$ given by and map $Y_4:\mathbb{C}^6\rightarrow \mathbb{C}^6$ given by are noninvolutive parametric Yang--Baxter maps which have a common Lax representation Lax-NLS. Moreover, map $Y_3$ has the functionally independent first integrals while map $Y_4$ admits the functionally independent first integrals

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Theorem 4.5
  • proof
  • Theorem 5.1
  • ...and 2 more