  Numerical simulations of domain wall propagation in thin nanotubes when an
external magnetic field is applied along the nanotube axis have shown an
unexpected behavior described as a transition from a linear to a magnonic
regime. As the applied magnetic field increases, the initial regime of linear
growth of the speed with the field is followed by a sudden change in slope
accompanied by the emission of spin waves. In this work an analytical formula
for the speed of the domain wall that explains this behavior is derived by
means of an asymptotic study of the Landau Lifshitz Gilbert equation for thin
nanotubes. We show that the dynamics can be reduced to a one dimensional
hyperbolic reaction diffusion equation, namely, the damped double Sine Gordon
equation, which shows the transition to the magnonic regime as the domain wall
speed approaches the speed of spin waves. This equation has been previously
found to describe domain wall propagation in weak ferromagnets with the
mobility proportional to the Dzyaloshinskii-Moriya interaction constant, for
Permalloy nanotubes the mobility is proportional to the nanotube radius.
