We use the nonlinear steepest descent method in the framework of the sine-Gordon model to study the behavior of dispersive activation and gapless waves at large times in a stripe domain structure of magnets and the nonadiabatic wave interaction with solitons in the domain structure. We show that the nonlinear interference of solitons and waves leads to oscillations of the soliton cores. Over time, they relax according to a power law. We determine the changes in the velocity and frequencies of solitons in a domain structure under the influence of spin waves.