Nonlinear dynamics of vortices in high-J(c) overdamped Josephson conta
cts with time-dependent transport current J(t) is considered. It is sh
own that equations of nonlocal Josephson electrodynamics which describ
e an Abrikosov vortex with Josephson core can be solved exactly for an
y J(t). Different nonlinear regimes, such as vortex acceleration after
a stepwise change of J(t) and oscillations of a vortex for periodic A
C currents are calculated analytically.