Asymptotic stability of Bernstein-Greene-Kruskal (BGK) waves, Landau Damping, and scattering theory

Abstract

We consider the two species (electrons and positive charged ions) Vlasov-Poisson system linearized around BGK waves. We formulate the problem as an equivalent Vlasov-Amp\`ere system, that we write as a system of Schr\"odinger type in an appropriate Hilbert space, and with a selfadjoint Vlasov-Amp\`ere operator as a Hamiltonian. We develop a complete stationary scattering theory. We identify the absolutely continuous spectrum and the singular spectrum of the Vlasov-Amp\`ere operator, we construct the generalized Fourier maps, we prove that the wave operators exist, are complete, satisfy Birman's invariance principle, and that the stationary formulae hold. Using these results we prove that the BGK waves are asymptotically stable. We obtain a precise description of the large time behaviour of the solutions to the Vlasov-Amp\`ere system Namely,for large times the phase-space densities of electrons and ions are asymptotic to the phase-space densities of solutions of the unperturbed Vlasov-Amp\`ere system. This implies that they follow the trajectories of solutions to Newton's equations for electrons and ions with the potential of the BGK wave, in the sense that they are transported along these trajectories. Furthermore, we prove that Landau damping holds, that is to say, the electric field tends to zero in pointwise sense for large times.