Collective coordinate study of spin wave emission from dynamic domain wall

We study theoretically the spin wave emission from a moving domain wall in a ferromagnet. Introducing a deformation mode describing a modulation of the wall thickness in the collective coordinate description, we show that thickness variation couples to the spin wave linearly and induces spin wave emission. The dominant emitted spin wave turns out to be polarized in the out-of wall plane ($\phi$)-direction. The emission contributes to the Gilbert damping parameter proportional to $\hbar\omega_\phi/K$, the ratio of the angular frequency $\omega_\phi$ of $\phi$ and the easy-axis anisotropy energy $K$.