  We present theoretical and numerical investigations of vortices driven by
strong dc and ac currents in long Josephson junctions described by a nonlinear
integro-differential equation which takes into account nonlocal electrodynamics
of films, vortex bremsstrahlung and Cherenkov radiation amplified by the
attraction of vortices to the edges of the junction. The work focuses on the
dynamics of vortices in Josephson junctions in thin films where the effects of
Josephson nonlocality dominate but London screening is negligible. We obtained
an exact solution for a vortex driven by an arbitrary time-dependent current in
an overdamped junction where the vortex turns into a phase slip if the length
of the junction is shorter than a critical length which depends on current. Our
analytical and numerical results show that the dynamic behavior of vortices
depends crucially on the ohmic damping parameter. In overdamped junctions
vortices expand as they move faster and turn into phase slips as current
increases. In underdamped junctions vortices entering from the edges produce
Cherenkov radiation generating cascades of expanding vortex-antivortex pairs,
which ultimately drive the entire junction into a resistive phase slip state.
Simulations revealed a variety of complex dynamic states of vortices under dc
and ac currents which can manifest themselves in hysteretic current-voltage
characteristics with jumps and regions with negative differential resistance
resulting from transitions from oscillating to ballistic propagation of
vortices, their interaction with pinning centers and standing nonlinear waves
in the junction.
