Large-amplitude electron plasma oscillations

Large-amplitude electron plasma oscillations in a one-dimensional, cold electron plasma are investigated for a spatially varying, immobile ion density. The Eulerian variables are transformed to Lagrangian variables. The problem is then treated as all effective one-dimensional particle in a potential. Tow examples of spatially varying ion densities that lead to analytic functions for the electron position, velocity and electric field are found by Gauss' law to have zero electron density, in unphysical result. A generic solution that includes the two examples is shown to have a spatially homogeneous ion density. A nonlinear ordinary differential equation is the condition for the appropriate form of the electron density and is solved by Lie group symmetries. A more general form of a solution is presented that possesses a spatially varying ion density, but when the necessary conditions are specified it has either zero electron density or secular terms in the electron density.