The analytic theory of 4 pi-kinks in long Josephson junctions is presented.
The laws for the velocity "quantized" spectrum of the different modes of t
he moving 4 pi-kinks nonlinear vortices are obtained. The discreteness of t
he velocity spectrum is dictated by the condition of the phase matching of
the nonlinear state which does not permit breaking of the vortices due to t
he Cherenkov radiation. (C) 1999 Elsevier Science B.V.