  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.
