Harmonic generation and departure from quasineutrality in ion sound and Langmuir wave coupling

The validity conditions of the Zakharov equations are reconsidered by investigating the linear stability analysis of a dipolar Langmuir wave. It is shown that the corresponding dispersion relation has a wider range of applicability, W/NT < (kλ_D)^-2, than the usual domain of validity of the Zakharov equations, W/NT < 1 (W/NT is the ratio of the Langmuir wave energy to the particle thermal energy, k is the characteristic wave number of the low-frequency perturbation). This result follows from an exact cancellation between the corrections as a result of the departure from quasineutrality and the contributions from the harmonics of the Langmuir wave. Such a cancellation is interpreted in terms of absence of nonlinear frequency shift caused by a dipolar pump wave in the dispersion relation of the Langmuir waves. The next-order corrections to the linearized Zakharov equations are computed; the proper renormalization of the low-frequency part of the dispersion relation is shown to result from the slow time variation and finite temperature corrections in the coupling constants.