Analytical solutions of Skyrme model

Man An,Nan Liu,Ruifeng Zhang

doi:10.3934/dcdss.2016092

Abstract

Exact analytic solutions of the kink soliton equation obtained ina recent interesting study of the classical Skyrme model definedon a simple spherically symmetric background are presented.By a variational method, the existence of spherically symmetric monopole solutions are proved. In particular, all finite-energy kink solitons must be Bogomool'nyi--Prasad--Sommerfield are showed. Moreover, together with numerical analysis, we can clearly see the validity of our theoretical results.

Highlights

Skyrme’s theory [21] is an important model in particle physics
A extraordinary property of Skyrme term is that excitations around Skyrme solitons may represent Fermionic degrees of freedom that could describe nucleons [2, 22]
The Skyrmions are the important objects which appear in many branches of physics such as astrophysics [19], condensed-matter physics [12], nematic liquids [11] and magnetics structures [18]

Summary

Introduction

Skyrme’s theory [21] is an important model in particle physics. A extraordinary property of Skyrme term is that excitations around Skyrme solitons may represent Fermionic degrees of freedom that could describe nucleons [2, 22]. The best known one for Skyrme models is the hedgehog ansatz for spherically symmetric systems, which reduces the field equations to a single scalar equation. In those studies, one mostly relies on numerical analyses, until very recently no exact analytic solutions of the Skyrme model with non-trivial topological charges were known. Exact spherically symmetric solutions of the Skyrme model with both a non-trivial winding number and a finite soliton mass (topological charge) are studied. We obtain the sharp properties of all finite-energy solutions of the Skyrme model within the Canfora, et al kink ansatz [7]. Reduce to the single ordinary differential equation for the Skyrmion profile α (6)

The reduced energy density is

The admissible space A is defined by

Or we have