We use the Riemann problem on a torus to obtain and analyze new analytic solutions of the Landau-Lifshitz model that describe the nonlinear dynamics of solitons of a biaxial ferromagnet in the field of dispersive spin waves. We show that nonlinear interference of solitons and waves leads to nonadiabatic relaxation oscillations of solitons. Formulas are obtained that determine the changes in the frequency and velocity of solitons in the radiation field. Collisions of relaxing solitons on a spin-wave background are analyzed.