Hierarchy of equations of motion for nonlinear coherent excitations applied to magnetic vortices

The shape of a solitonlike excitation in a nonintegrable system generally depends on the velocity and all higher-order time derivatives of the position X→ of the excitation. Using a sequence of generalized traveling-wave Ans\"atze we derive a hierarchy of equations of motion for X→. The type of excitation determines on which levels the hierarchy can be truncated consistently: ``Gyrotropic'' excitations are governed by odd-order equations, nongyrotropic ones by even-order equations. Examples for the latter case are kinks in one-dimensional models and planar vortices of the two-dimensional anisotropic (easy-plane) Heisenberg model. The nonplanar vortices of this model are the simplest gyrotropic example. For this last case we solve the Hamilton equations for a finite system with one vortex and free boundary conditions and calculate the parameters of the third-order equation of motion. This equation yields trajectories which are a superposition of two cycloids with different frequencies, which is in full agreement with computer simulations of the full many-spin model. Finally we demonstrate that the additional effects from the fifth-order equation are negligible.