Exact knot solutions in a generalized Skyrme-Faddeev model

TL;DR

This paper introduces a generalized Skyrme-Faddeev model with an extra scalar field, finding exact and numerical knot solutions with implications for understanding topological solitons and their energies.

Contribution

It presents a new generalized model admitting exact knot solutions and provides numerical analysis for higher charges, improving understanding of knot energies in such theories.

Findings

Exact toroidal solitons exist in a special parameter case.

Numerical solutions for higher Hopf charges are obtained.

Knot energies follow the E_H ≈ Q_H^{3/4} law.

Abstract

We propose a generalized Skyrme-Faddeev type theory with an additional scalar field. In a special case of model parameters one has a theory which admits exact knot solutions given by a class of exact toroidal solitons from Aratyn-Ferreira-Zimerman (AFZ) integrable CP1 model. In a general case the theory admits an exact knot solution for a unit Hopf charge. For higher Hopf charges we perform numeric analysis of the solutions and obtain estimates for the knot energies using energy minimization procedure based on ansatz with AFZ field configurations and with rational functions. We show that AFZ configurations provide a better approximate solutions. The corresponding knot energies are in a good agreement with a standard law for the low energy bound, E_H\simeq Q_H^{3/4}.