Investigation of the Nicole model

TL;DR

This paper investigates soliton solutions in the Nicole model, a non-linear field theory, revealing linear energy growth with topological charge for symmetric solutions and sub-linear bounds for the full system.

Contribution

It introduces a numerical and analytical study of solitons in the Nicole model, demonstrating the energy-charge relationship and bounds, and compares symmetric and full solutions.

Findings

Soliton energies grow linearly with Hopf index for symmetric solutions.

A sub-linear upper bound on energy growth is proven for the full system.

Symmetric solitons are not energy minimizers for large topological charges.

Abstract

We study soliton solutions of the Nicole model - a non-linear four-dimensional field theory consisting of the CP^1 Lagrangian density to the non-integer power 3/2 - using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behaviour to hold exactly for the ansatz. On the other hand, for the full three-dimensional system without symmetry reduction we prove a sub-linear upper bound, analogously to the case of the Faddeev-Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev-Niemi model.