Fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model

The scattering of Dirac fermions in the background fields of topological solitons of the $(2+1)$-dimensional nonlinear $O(3)$ $\sigma$-model is studied using both analytical and numerical methods. General formulae describing fermion scattering are obtained and the symmetry properties of the partial scattering amplitudes and elements of the $S$-matrix are determined. Within the framework of the Born approximation, the scattering amplitudes, differential cross-sections, and total cross-sections of fermion-soliton scattering are obtained in analytical forms, and their symmetry properties and asymptotic behavior are investigated. The dependences of the first several partial elements of the $S$-matrix on the momentum of the fermion are obtained using numerical methods, and some properties of these dependences are ascertained and discussed.