Geodesic
Axially-symmetric topological configurations in the Skyrme and Faddeev chiral models
Eurasian mathematical journal, Tome 6 (2015) no. 2, pp. 82-89.

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By definition, in chiral model the field takes values in some homogeneous space $G/H$. For example, in the Skyrme model (SM) the field is given by the unitary matrix $U\in SU(2)$, and in the Faddeev model (FM) — by the unit $3$-vector $\mathbf{n}\in S^2$. Physically interesting configurations in chiral models are endowed with nontrivial topological invariants (charges) $Q$ taking integer values and serving as generators of corresponding homotopic groups. For SM $Q=\mathrm{deg}(S^3\to S^3)$ and is interpreted as the baryon charge $B$. For FM it coincides with the Hopf invariant $Q_H$ of the map $S^3\to S^2$ and is interpreted as the lepton charge. The energy $E$ in SM and FM is estimated from below by some powers of charges: $E_S>\mathrm{const|Q|}$, $E_F>\mathrm{const}|Q_H|^{3/4}$. We consider static axially-symmetric topological configurations in these models realizing the minimal values of energy in some homotopic classes. As is well-known, for $Q=1$ in SM the absolute minimum of energy is attained by the so-called hedgehog ansatz (Skyrmion): $U=\exp[i\Theta(r)\sigma]$, $\sigma=(\sigma\mathbf{r})/r$, $r = |\mathbf{r}|$, where $\sigma$ stands for Pauli matrices. We prove via the variational method the existence of axially-symmetric configurations (torons) in SM with $|Q|>1$ and in FM with $|Q_H|\geqslant1$, the corresponding minimizing sequences being constructed, with the property of weak convergence in $W_\infty^1$. 
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     title = {Axially-symmetric topological configurations in the {Skyrme} and {Faddeev} chiral models},
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Yu. P. Rybakov. Axially-symmetric topological configurations in the Skyrme and Faddeev chiral models. Eurasian mathematical journal, Tome 6 (2015) no. 2, pp. 82-89. http://geodesic.mathdoc.fr/item/EMJ_2015_6_2_a5/

[1] I. Ekland, R. Temam, Convex analysis and variational problems, North-Holland Publishing Company, Amsterdam, 1976 | MR

[2] K. Fujii, S. Otsuki, F. Toyoda, “A soliton solution with baryon number $B=0$ and Skyrmion”, Progr. Theor. Phys., 73:2 (1985), 524–527 | DOI | MR

[3] V. G. Makhankov, Yu. P. Rybakov, V. I. Sanyuk, The Skyrme model. Fundamentals, methods, applications, Springer-Verlag, Berlin, 1993 | MR

[4] J. W. Milnor, J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, New Jersey, 1974 | MR | Zbl

[5] H. K. Moffatt, “The degree of knotteddness of tangled vortex lines”, J. Fluid Mech., 35, Part I (1969), 117–129 | DOI | Zbl

[6] R. Palais, “The principle of symmetric criticality”, Comm. Math. Phys., 69:1 (1979), 19–30 | DOI | MR | Zbl

[7] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, New Jersey, 1951 | MR | Zbl

[8] Yu. P. Rybakov, “Stability of many-dimensional solitons in chiral models and gravity”, Gravity and cosmology, Itogi Nauki i Tekhniki. Series “Classical field theory and gravitational theory”, 2, VINITI, M., 1991, 56–111 (in Russian)

[9] T. H. R. Skyrme, “A unified field theory of mesons and baryons”, Nucl. Phys., 31:4 (1962), 556–569 | DOI | MR

[10] A. F. Vakulenko, L. V. Kapitansky, “Stability of solitons in $S^2$ nonlinear sigma-model”, Doklady of Ac. Sci. USSR, 146:4 (1979), 840–842 (in Russian) | MR