Mathematics

Electron plasma waves were first introduced nearly 90 years ago, in the celebrated paper by Tonks and Langmuir [1]. Since then, they have been the subject of a countless number of papers, leading to unexpected results, such as their collisionless damping, first described by Landau [2]. Yet, no rigorous description of their nonlinear propagation has been derived, even within the simple limit of geometrical optics.The talk aims at filling this gap by addressing the issue of nearly monochromatic plasma waves in the strongly nonlinear regime, when a significant fraction of electrons have been trapped in the potential trough. The waves may either result from a purely electrostatic instability or may be externally driven, for example with lasers. The latter situation is particularly important for laser fusion, since driven electrostatic waves may induce a large reflectivity of the incoming laser energy through stimulated Raman scattering. Starting directly from the Vlasov-Gauss system, it will be shown that the wave propagation can be modeled with a simple envelope equation, which does account for nonlinear kinetic effects. In particular, collisionless dissipation will be described in a regime when Landau’s theory (or its nonlinear counterpart by Mouhot and Villani [3]) does not apply. A very simple derivation for what may be viewed as the nonlinear Landau damping rate of a driven plasma wave will be provided in a one-dimensional and a three-dimensional geometry. Conversely, when the wave spontaneously grows unstable, the nonlinear growth and saturation of the instability will be derived analytically. The key point of the theory rests in an approximate resolution of Vlasov equation, obtained by matching two distinct perturbative approaches : one in the smallness of the wave amplitude, and one in the slowness of its variations. Moreover, the results from the perturbative analyses are used within a nonlocal variational formalism [4] in order to account for plasma inhomogeneity and non-stationarity.References[1] Lewi Tonks and Irving Langmuir, Phys. Rev. 13, 195 (1929)[2] Lev Davidovitch Landau, J. Phys. (Moscow)10, 25 (1946).[3] C. Mouhot and C. Villani, Acta Math. 207, 29 (2011).   [4] Didier Bénisti, Phys. Plasmas 23, 102015 (2016).