Instability conditions for some periodic BGK waves in the Vlasov-Poisson system

Abstract

A one-dimensional, collisionless plasma given by the Vlasov-Poisson system is considered and the stability properties of periodic steady state solutions known as Bernstein-Greene-Kruskal (BGK) waves are investigated. Sufficient conditions are determined under which BGK waves are linearly unstable under perturbations that share the same period as the equilibria. It is also shown that such solutions cannot support a monotonically decreasing particle distribution function.PACS. 52.25.Dg Plasma kinetic equations -02.30.Jr Partial differential equations -52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams

Trapping scaling for bifurcations in the Vlasov systems

“…In the plasma physics context, Bernstein-Greene-Kruskal exhibited a large family of exact solutions of the Vlasov-Poisson equation [13], now commonly called BGK modes. Some of these solutions are stable, others unstable [14][15][16][17]. The non linear behavior in case of instability is an open question.…”

Towards a classification of bifurcations in Vlasov equations

“…The simplest cases, based on the one-dimensional Vlasov-Poisson equation, are similar to the HMF example studied above, and we know that bifurcations do occur (see, for instance, Ref. [38]): we expect that when a BGK mode becomes unstable, the behavior of the instability may, at least in some instance, be described by the theory put forward in this paper.…”

“…We will show that the linearized Vlasov equation around a stationary state has a size-3 Jordan block structure (41), and give the expressions of the generalized eigenvectors and eigenprojections at the critical point [Eqs. (36)- (38) and (45) for the HMF case]. Note that, in this Appendix, we will omit the upper tilde on Fourier components for simplicity of notation, while the subscript 0 of F 0 represents the critical point μ = 0.…”

Towards a classification of bifurcations in Vlasov equations

“…Instabilities of Bernstein-Greene-Kruskal modes in plasmas provide a vast class of natural candidates. The simplest cases, based on the 1D Vlasov-Poisson equation, are similar to the HMF example studied above, and we know that bifurcations do occur (see for instance [31]): we expect some of these bifurcations to be described by the theory put forward in this paper. Radial orbit instability is well known in astrophysics (see for instance [32]), and believed to play a role in determining the structure of galaxies.…”