The Faddeev knots as stable solitons: existence theorems
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.
scientific article; zbMATH DE number 2170597
scientific article