  We study vortex dynamics in trapped two-component Bose-Einstein condensates
with a laser- induced spin-orbit coupling using the numerical analysis of the
Gross-Pitaevskii equation. The spin-orbit coupling leads to three distinct
ground state phases, which depend on some experimentally controllable
parameters. When a vortex is put in one or both of the two-component
condensates, the vortex dynamics exhibits very different behaviors in each
phase, which can be observed in experiments. These dynamical behaviors can be
understood by clarifying the stable vortex structure realized in each phase.
