  The motion of a vortex domain wall in a ferromagnetic strip of submicron
width under the influence of an external magnetic field exhibits three distinct
dynamical regimes. In a viscous regime at low fields the wall moves rigidly
with a velocity proportional to the field. Above a critical field the viscous
motion breaks down giving way to oscillations accompanied by a slow drift of
the wall. At still higher fields the drift velocity starts rising with the
field again but with a much lower mobility dv/dH than in the viscous regime. To
describe the dynamics of the wall we use the method of collective coordinates
that focuses on soft modes of the system. By retaining two soft modes,
parametrized by the coordinates of the vortex core, we obtain a simple
description of the wall dynamics at low and intermediate applied fields that
describes both the viscous and oscillatory regimes below and above the
breakdown. The calculated dynamics agrees well with micromagnetic simulations
at low and intermediate values of the driving field. In higher fields,
additional modes become soft and the two-mode approximation is no longer
sufficient. We explain some of the significant features of vortex domain wall
motion in high fields through the inclusion of additional modes associated with
the half-antivortices on the strip edge.
