Modified Poisson eigenfunctions for electrostatic Bernstein–Greene–Kruskal equilibria

Abstract

The stability of an electrostatic Bernstein–Greene–Kruskal equilibrium by Lewis and Symon’s general linear stability analysis for spatially inhomogeneous Vlasov equilibria, which employs eigenfunctions and eigenvalues of the equilibrium Liouville operator and the modified Poisson operator, is considered. Analytic expressions for the Liouville eigenfuctions and eigenvalues have already been given; approximate analytic expressions for the dominant eigenfunction and eigenvalue of the modified Poisson operator are given. In the kinetic limit three methods are given: (i) the perturbation method, (ii) the Rayleigh–Ritz method, and (iii) a method based on a Hill’s equation. In the fluid limit the Rayleigh–Ritz method is used. The dominant eigenfunction and eigenvalue are then substituted in the dispersion relation and the growth rate calculated. The growth rate agrees very well with previous results found by numerical simulation and by modified Poisson eigenfunctions calculated numerically.