  We analyze the dynamics of Josephson vortex states in two-component
Bose-Einstein condensates with Rashba-Dresselhaus spin-orbit coupling by using
the Gross-Pitaevskii equation. In 1D, both in homogeneous and harmonically
trapped systems, we report on stationary states containing doubly charged,
static Josephson vortices. In multidimensional systems, we find stable
Josephson vortices in a regime of parameters typical of current experiments
with $^{87}$Rb atoms. In addition, we discuss the instability regime of
Josephson vortices in disk-shaped condensates, where the snake instability
operates and vortex dipoles emerge. We study the rich dynamics that they
exhibit in different regimes of the spin-orbit coupled condensate depending on
the orientation of the Josephson vortices.
