Nonlinear Schr\"odinger type tetrahedron maps

TL;DR

This paper develops new solutions to the functional and parametric tetrahedron equations using Darboux transformations related to NLS and DNLS equations, resulting in novel high-dimensional tetrahedron maps with potential applications in integrable systems.

Contribution

It introduces a method for constructing solutions to the tetrahedron equations via Darboux transformations, including new nine-dimensional and six-dimensional tetrahedron maps.

Findings

Constructed nine-dimensional tetrahedron maps from Darboux transformations.

Restricted nine-dimensional maps to six-dimensional invariant leaves.

Developed parametric tetrahedron maps using degenerated Darboux transformations.

Abstract

This paper is concerned with the construction of new solutions in terms of birational maps to the functional tetrahedron equation and parametric tetrahedron equation. We present a method for constructing solutions to the parametric tetrahedron equation via Darboux transformations. In particular, we study matrix refactorisation problems for Darboux transformations associated with the nonlinear Schr\"odinger (NLS) and the derivative nonlinear Schr\"odinger (DNLS) equation, and we construct novel nine-dimensional tetrahedron maps. We show that the latter can be restricted to six-dimensional parametric tetrahedron maps on invariant leaves. Finally, we construct parametric tetrahedron maps employing degenerated Darboux transformations of NLS and DNLS type.