The Faddeev knots as stable solitons: existence theorems
type: Item
instance of: scholarly article
title: The Faddeev knots as stable solitons: existence theorems
zbMATH Open document ID: 1087.35082
author: Yisong Yang
author: Fanghua Lin
published in: Science in China. Series A
publication date: 2005-05-23T00:00:00Z
review text: In the 70th of the last century L. Faddeev proposed a model of solitons localized around one-dimensional contours of string type: circle, knot and link. Their existence was certified by Faddeev, Niemi, Battye and Sutkliffe, Hietarinta and Salo (1997--1999), who performed computer simulation for the solution configurations of suitable Lagrangian field theory equations with the Hopf charge \(Q=1,2,\dots,8\). They found that for \(Q \leq 5\) the energy-minimizing solitons are ring-shaped and higher charges cause greater distortion and the solitons become knotted and more complex. However, any theoretical investigations of the problem were absent. In the paper under review, the authors give the sufficiently full list of preceeding publications of many authors which in some manner may be connected to the problem and then announce a series of existence theorems for the Faddeev stable knotted solitons realized as the static energy minimizers of the energy in the strata characterized by the Hopf invariant. In the main part of the paper the authors recall the basic definitions of the Faddeev model: action in the Minkovski space with signature \((+---)\), energy \(E\), Hopf charge and Hopf invariant \(Q(n)\), where \(n\) denotes the field value on the unit 2-sphere. The existence theorems concern the cases of the entire space and of a bounded domain. The most interesting results are related to the first case: there are the existence of a least-energy Faddeev soliton and the existence of energy minimizers among infinitely many topological classes. And the sublinear asymptotic estimate \(E \simeq |Q|^{3/4}\) is of great interest, since it implies that a knotted (clustered) soliton structure is preferred over a separated multisoliton structure. This excellent paper permits to think that the problems arised after numerical investigation of the Faddeev model may be solved in future. The authors promise a future more detailed publication.
reviewed by: Valentina Golubeva
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 57T20
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 2170597
zbMATH Keywords: Faddeev solitons model Hopf invariant
zbMATH Keywords: energy minimizer
zbMATH Keywords: knotted solitons
zbMATH Keywords: existence
zbMATH Keywords: sublinear asymptotic energy estimation
zbMATH Keywords: least-energy Faddeev soliton
zbMATH Keywords: knotted soliton structure
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2170597