Nonlinear frequency shift, wave breaking and saturation via sideband instability of electron plasma waves in the adiabatic approximation.

Electrons trapped in the wave troughs often lead to plasma instabilities, either self-consistent such as the beam plasma instability or externally driven such as stimulated Raman scattering. These instabilities generally arise at wave amplitudes large enough so that common linearization techniques do not give quantitative results anymore. It is what we call the nonlinear regime.

Here we will show a method to self consistently compute the electron distribution function in the nonlinear regime, when the nonlinearity is due to electron trapping in the wave troughs. This method applies whatever the variations of the wave amplitude, provided that these variations are slow enough and is valid even when the plasma is inhomogeneous and non stationary. This requires a self-consistent description of the electron plasma wave scalar and vector potentials as well as of the nonlinear frequency. It accounts for various nonlinear effects such as the change in phase velocity making the wave frame non-inertial, the transition probabilities from one region of phase space to the other when an orbit crosses the separatrix, the possible change in direction of the wavenumber and others. The relative importance of each of these effects will be discussed in detail and a discussion of the wave breaking because of the impossibility to solve the dispersion relation above certain amplitude presented.

A validation against numerical simulations by test particles in order to check self-consistency and comparison with previous results will be provided.

The obtained distribution function will then be propagated in time by a Cheng and Knorr Vlasov-Poisson solver with cubic spline interpolation in order to observe the growth of sidebands. A theoretical method for computing the growth rate of these sidebands will be given and compared with previously obtained results such as those of Kruer or Dodin.