Nonlinear wave damping on an electron hole in a weakly inhomogeneous plasma

Nonlinear damping of the Langmuir wave in a weakly inhomogeneous plasma with a negative concentration gradient is considered. For phase velocities close to the thermal velocity of electrons, their dispersion strongly differs from the linear one and cannot be taken into account by means of a small correction to the linear dispersion law. Based on the energy balance and nonlinear dispersion equations, a dependence of the wave amplitude on its phase velocity is established together with the limiting phase velocity at which the wave is completely damped.