A first integration of some knot soliton models

TL;DR

This paper demonstrates that certain knot soliton models, including the Nicole and Aratyn-Ferreira-Zimerman models, contain sectors where equations simplify to first order and include known topologically nontrivial solutions.

Contribution

It identifies and analyzes sectors within these models that admit first order equations and contain all known Hopf solitons, advancing understanding of their topological properties.

Findings

Sectors with first order equations exist in these models.

All known Hopf solitons are contained in these sectors.

These sectors are topologically nontrivial.

Abstract

Recently it has been shown that there exists a sector within the Faddeev-Niemi model for which the equations of motion may be reduced to first order equations. However, no solutions to that sector have been given. It is not even known whether this sector contains topologically nontrivial solutions, at all. Here, we show that two models with analytically known Hopf solitons, namely the Nicole and the Aratyn-Ferreira-Zimerman models, possess sectors which can be integrated to first order partial differential equations. The main result is that these sectors are topologically nontrivial. In fact, all analytically known hopfions belong to them.