Axially-symmetric topological configurations in the Skyrme and Faddeev chiral models

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 $W_\infty^1$.