Sectionally Analytic Distribution Functions of Electron and Ions associated with a BGK wave in a Collisionless Plasma

Abstract

In this work, we invetigate further the singular nature of the nonlinear stationary solutions of the one dimensional Vlasov-Poisson system of equations, which governs a plasma made of electron and one species of fully ionised ions. First, we propose a new integral formulation of the Poisson equation and we prove two inversion lemmas for such equation. These lemmas allow us to write the solutions of the Poisson equation in such a way that the energy distribution of either of the particle species is related, in a straightforward way, to the energy distribution of the other species. Then, we show that these distribution functions are retrieved as boundary values of suitable sectionally analytic functions. These latter functions are shown tobe the extension of the particle distributions into their respective complex energy domai