Annihilation of Domain Walls in a Ferromagnetic Wire

We study the annihilation of topological solitons in the simplest setting: a one-dimensional ferromagnet with an easy axis. We develop an effective theory of the annihilation process in terms of four collective coordinates: two zero modes of the translational and rotational symmetries $Z$ and $\Phi$, representing the average position and azimuthal angle of the two solitons, and two conserved momenta $\zeta$ and $\varphi$, representing the relative distance and twist. Comparison with micromagnetic simulations shows that our approach captures well the essential physics of the process.