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

TL;DR

This paper develops a hierarchy of equations of motion for nonlinear excitations like magnetic vortices, showing how their dynamics depend on derivatives of their position and validating the approach with simulations.

Contribution

It introduces a hierarchy of equations of motion for nonlinear excitations, linking excitation type to the order of the governing equations, and applies it to magnetic vortices.

Findings

Derived a hierarchy of equations of motion for excitations.

Validated the third-order equation with simulations.

Found higher-order effects are negligible.

Abstract

Starting from a travelling wave ansatz we show successively that the shape of a nonlinear excitation generally depends also on the 1st, 2nd, ... time derivative of the position X of the excitation. From the Hamilton equations we derive a hierarchy of equations of motion for X. The type of the excitation determines on which levels the hierarchy can be truncated consistently: "Gyrotropic" excitations are governed by odd-order equations, non-gyrotropic ones by even-order equations. Examples for the latter case are kinks in 1-dimensional models and planar vortices of the 2D anisotropic (easy-plane) Heisenberg model. The non-planar vortices of this model are the simplest gyrotropic example. For this case we solve the Hamilton equations for a finite system with one vortex and free boundary conditions and calculate the parameters of the 3rd-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 5th-order equation are negligible.