  We investigate the phase dynamics of a long Josephson junction (LJJ) with
two-gap superconductors. In this junction, two channels for tunneling between
the adjacent superconductor (S) layers as well as one interband channel within
each S layer are available for a Cooper pair. Due to the interplay between the
conventional and interband Josephson effects, the LJJ can exhibit unusual phase
dynamics. Accounting for excitation of a stable 2$\pi$-phase texture arising
from the interband Josephson effect, we find that the critical current between
the S layers may become both spatially and temporally modulated. The spatial
critical current modulation behaves as either a potential well or barrier,
depending on the symmetry of superconducting order parameter, and modifies the
Josephson vortex trajectories. We find that these changes in phase dynamics
result in emission of electromagnetic waves as the Josephson vortex passes
through the region of the 2$\pi$-phase texture. We discuss the effects of this
radiation emission on the current-voltage characteristics of the junction.
