Sideband growth rates for differentiable and discontinuous distribution functions

This article addresses the stability of a nonlinear electron plasma wave (EPW) against the growth of longitudinal sidebands. The electron distribution function consistent with the EPW is assumed to only depend on the dynamical action. Consequently, the EPW is either stationary (a so-called Berstein-Greene-Kruskal mode) or varies very slowly in space and time (a so-called adiabatic wave). The sideband growth rates and the unstable spectrum are calculated theoretically by accounting for the exact nonlinear electron orbits in the EPW. Our theoretical results are compared against those from previous theories, and also against those from Vlasov simulations when the distribution function is differentiable. The latter comparisons show that our theory may also apply when the electron distribution function depends on, both the action and the angle. Moreover, our theory allows for discontinuous distributions, which are consistent with important classes of EPWs, e.g., those resulting from stimulated Raman scattering. Addressing such distributions using kinetic codes remains a challenge. Nevertheless, using our theoretical results, in this article we discuss the impact of discontinuities on the unstable spectrum in the linear regime, and on the range of validity of the linear approximation.