Internal modes and magnon scattering on topological solitons in two-dimensional easy-axis ferromagnets

We study the magnon modes in the presence of a topological soliton in a two-dimensional Heisenberg easy-axis ferromagnet. The problem of magnon scattering on the soliton with arbitrary relation between the soliton radius R and the “magnetic length” ${\Delta }_0$ is investigated for partial modes with different values of the azimuthal quantum numbers m. Truly local modes are shown to be present for all values of m, when the soliton radius is large enough. The eigenfrequencies of such internal modes are calculated analytically on the limiting case of a large soliton radius and numerically for arbitrary soliton radius. It is demonstrated that the model of an isotropic magnet, which admits an exact analytical investigation, is not adequate even for the limit of small radius solitons, $R\ll {\Delta }_0:$ there exists a local mode with nonzero frequency. We use the data of local modes to derive the effective equation of soliton motion; this equation has the usual Newtonian form in contrast to the case of the easy-plane ferromagnet. The effective mass of the soliton is found.