  We report numerical simulations of the nonlinear dynamics of Josephson
vortices driven by strong dc currents in layered superconductors. Dynamic
equations for interlayer phase differences in a stack of coupled
superconducting layers were solved to calculate a drag coefficient $\eta(J)$ of
the vortex as a function of the perpendicular dc current density $J$. It is
shown that Cherenkov radiation produced by a moving vortex causes significant
radiation drag increasing $\eta(v)$ at high vortex velocities $v$ and striking
instabilities of driven Josephson vortices moving faster than a terminal
velocity $v_c$. The steady-state flux flow breaks down at $v>v_c$ as the vortex
starts producing a cascade of expanding vortex-antivortex pairs evolving into
either planar macrovortex structures or branching flux patterns propagating
both along and across the layers. The pair production triggered by a rapidly
moving vortex is most pronounced in a stack of underdamped planar junctions
where it can occur at $J>J_s$ well below the interlayer Josephson critical
current density. Both $v_c$ and $J_s$ were calculated as functions of the
quasiparticle damping parameter, and the dc magnetic field applied parallel to
the layers. The effects of vortex interaction on the Cherenkov instability of
moving vortex chains and lattices in annular stacks of Josephson junctions were
considered. It is shown that a vortex driven by a current density $J>J_s$ in a
multilayer of finite length excites self-sustained large-amplitude standing
waves of magnetic flux, resulting in temporal oscillations of the total
magnetic moment. We evaluated a contribution of this effect to the power $W$
radiated by the sample and showed that $W$ increases strongly as the number of
layers increases.
