Cross-fertilization of Ferreira's Hopfions And Electromagnetic Knots

TL;DR

This paper explores the relationship between Ferreira's Hopf solitons and electromagnetic knots, showing how each can be realized as solutions of a conformal nonlinear sigma model and comparing their energies and charge distributions.

Contribution

It demonstrates the mutual realization of Ferreira's Hopfions and electromagnetic knots within a conformal nonlinear sigma model, revealing new exact solutions and their physical properties.

Findings

Electromagnetic knots are exact solutions of the sigma model different from Ferreira's solutions.

Ferreira's solutions can realize magnetic knots.

The energy and charge distribution of these knots are analyzed.

Abstract

The interrelation between Ferreira's Hopf solitons of a conformal nonlinear $\sigma$ model and the electromagnetic knots found by Ra$\tilde{\rm{n}}$ada et al. is investigated. It is shown that the electromagnetic knots yield exact solutions of the conformal nonlinear $\sigma$ model different from those obtained by Ferreira. Conversely, It is discussed that Ferreira's solutions realize magnetic knots. The energy associated with these two kinds of knots are compared. The structure of the electric charge distribution and the electric current density associated with the magnetic knots is investigated.