Stability of small BGK waves

Bernstein–Greene–Kruskal (BGK) waves are fundamental inhomogeneous steady states of the one-dimensional Vlasov–Poisson system, introduced in 1957. Despite their classical role, their stability theory remains largely open. The only known result is by Guo-Lin, who considered a special class of small BGK waves. In this talk, I will present a stability–instability criterion for small BGK waves bifurcating from homogeneous equilibria, determined by the behavior of the particle distribution near the separatrix.  This is joint work with D. Bian, E. Grenier and B. Pausader.