A self-consistent model combining the kinetic Vlasov equation governing the distribution function of the resonant particles and the Ampere-Maxwell law is presented. The nonresonant particles are taken into account via the dielectric permittivity in the linear approximation. The results of numerical simulations demonstrating the nonlinear evolution of the broad spectrum of waves induced by an unstable distribution of electrons in homogeneous and inhomogeneous plasma are analyzed. The frequency-time analysis of the electric field reveals the presence of quasi-periodic elements with increasing frequency. It is demonstrated that temporal modulation of the amplitude of the electric field appears due to selection of harmonics with equally spaced wavenumbers in the initial spectrum of the waves. This choice of harmonics gives rise to spatial periodic structures of the electric field that transform into temporal modulation seen in the spectrograms upon propagation.
A.A. Luzhkovskiy (Wed,) studied this question.