  The 1D Vlasov-Poisson system is the simplest kinetic model for describing an
electrostatic collisonless plasma, and the BGK waves are its famous exact
steady solutions. They play an important role on the long time dynamics of a
collisionless plasma as potential "final states" or "attractors", thanks to
many numerical simulations and observations. Despite their importance, the
existence of stable BGK waves has been an open problem since their discovery in
1958. In this paper, first linearly stable BGK waves are constructed near
homogeneous states.
