Part IV. Applications to the Vlasov Plasma A Resonant Far Fields and Amplitude Oscillations

The present paper considers a one dimensional monochromatic Langmuir wave of small but finite amplitude. In §1, long·time asymptotic behaviors of the wave due to resonant effects are briefly discussed as the resonant far fields of the Vlasov equation. In §21 following the idea of the reductive perturbation method the Vlasov-Poisson equations are investigated by Lagrangian method assuming appropriate orderings to give a set of equations of one slow time. The expansion parameter e is the ratio of the decay constant of the Landau damping YL to the plasma frequency, wp, i.e., The resultant set of equations contains a parameter q=.yL.'ws, with ws, the bounce frequency of electron. When q>1 (but q<e-1), the wave Landau-damps, in which case trapped electrons are not essential to the time evolution of the wave amplitude E. When q<l, a density perturbation of trapped electrons balances with a collective slow variation of E, and a system of equations which governs the time evolution of E is obtained and integrated numerically. An initial evolution of this system is solved analytically and a nonlinear corrections to the Landau damping is obtained. The results of numerical investigations of the system for all time given in §3 are as follows. For q<0.77 (but q>e), in the time dependence of the amplitud