Langmuir wave in weakly inhomogeneous relativistic plasma

The evolution of a Langmuir wave in a weakly inhomogeneous relativistic plasma with a positive density gradient is considered. It is shown that, at relativistic phase velocities, the wave evolution even at the tail of the electron distribution, where it is close to linear in the nonrelativistic case, results in the wave transformation into a hybrid of two waves with different spatial periods. Nonlinear dispersion relations for different stages of the wave evolution are derived.