  In this paper, we address the theoretical resolution of the Vlasov-Gauss
system from the linear regime to the strongly nonlinear one, when significant
trapping has occurred. The electric field is that of a sinusoidal electron
plasma wave (EPW) which is assumed to grow from the noise level, and to keep
growing at least up to the amplitude when linear theory in no longer valid
(while the wave evolution in the nonlinear regime may be arbitrary). The ions
are considered as a neutralizing fluid, while the electron response to the wave
is derived by matching two different techniques. We make use of a perturbation
analysis similar to that introduced to prove the Kolmogorov-Arnold- Moser
theorem, up to amplitudes large enough for neo-adiabatic results to be valid.
Our theory is applied to the growth and saturation of the beam-plasma
instability, and to the three-dimensional propagation of a driven EPW in a
non-uniform and non-stationary plasma. For the latter example, we lay a special
emphasis on nonlinear collisionless dissipation. We provide an explicit
theoretical expression for the nonlinear Landau-like damping rate which, in
some instances, is amenable to a simple analytic formula. We also insist on the
irreversible evolution of the electron distribution function, which is nonlocal
in the wave amplitude and phase velocity. This makes trapping an effective
means of dissipation for the electrostatic energy, and also makes the wave
dispersion relation nonlocal. Our theory is generalized to allow for stimulated
Raman scattering, which we address up to saturation by accounting for plasma
inhomogeneity and non-stationarity, nonlinear kinetic effects, and interspeckle
coupling.
