Stability of Bernstein–Greene–Kruskal equilibria

The recent linear stability analysis of Lewis and Symon for spatially inhomogeneous Vlasov equilibria is illustrated with the case of an unstable periodic wavetrain of strongly inhomogeneous Bernstein–Greene–Kruskal equilibria. The stability formalism involves expanding an auxiliary function related to the perturbation distribution function in terms of the equilibrium Liouville eigenfunctions, and expanding the perturbation potential in terms of the eigenfunctions of an appropriately chosen field operator. The infinite-dimensional dispersion matrix is truncated to M×M by assuming that the normal mode of interest of the perturbation potential can be adequately represented by M eigenfunctions of the field operator; the eigenfrequencies ω are the zeroes of the determinant of the dispersion matrix. A particular Bernstein–Greene–Kruskal equilibrium was chosen as a numerical example, and the growth rate and normal mode of the instability were determined by numerical simulation. The agreement of the theory with the simulation for the growth rate and normal mode of the instability was excellent, and it was possible to choose a field operator a priori such that a 1×1 dispersion matrix was sufficient for obtaining accurate results.