Second-order oscillations of a Vlasov–Poisson plasma in Fourier-transformed velocity space

Abstract

We study the Vlasov–Poisson system of equations in Fourier-transformed velocity space. First we reformulate some results of the linear theory: in the new representation the van Kampen–Case eigenmodes are found to be ordinary functions with convenient continuity properties. We give a transparent derivation of the free-streaming temporal echo in terms of the kinematics of wave packets in Fourier-transformed velocity space. We further extend this analysis to include Coulomb interactions, which allows us to establish a connection between the echo theory, the second-order oscillations of Best and the phenomenon of linear side bands. The calculation of the time evolution of the global second-order electric field is performed in detail in the case of a Maxwellian equilibrium distribution function. We conclude that the phenomenon of linear side bands may be properly explained in terms of the intrinsic features of the equilibrium distribution function.

Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium

“…The frequencies (I) and (II) have already been identified in our preliminary work on this subject [4], but not the frequency (III). We emphasize that all the three type of frequencies are listed in [13], which makes our analysis coherent with the physics litterature. To conclude this presentation, we are going to state a precise theorem giving the asymptotic behavior of the solutions of (VPL2).…”

“…By construction of g through Duhamel formula (20), g is obviously a continuous function on R + × T d × R d and C 1 on R * + × T d × R d . Furthermore, we may verify by a straightforward calculation that g is solution of the Vlasov equation (13). Consequently, we just have to prove that g, u is solution of Poisson equation (18).…”

“…In that case, we refer to Denavit [8], for one of the first works on the subject, in 1965. Different second order oscillations appear and have been studied by physicists (see for example [13] and references therein; there are many references especially from the 1960s and 1970s). Our aim is to make here a rigorous mathematical study of the asymptotical behavior of the solutions of these equations, which has, to the best of our knowledge, not already been performed.…”

Solar acoustic oscillations in a random density field

“…(9) and (11) be attenuated in time?" Our interpretation of this apparent paradox is that the root of the dispersion function which lies in the lower half of its unphysical Riemann sheet has no associated eigenfunction; the damped oscillation related to this root is rather similar to the "virtual modes" found in Landau damping (Sedláček & Nocera 1992), in inhomogeneous plasmas (Sedláček 1971(Sedláček , 1994 and in the inhomogeneous vibrating string (Sedláček et al 1986): they can be aptly named "random virtual modes".…”

Non-linear free streaming in Vlasov plasma

“…In our recent papers [1,2,3,4,5] we drew attention to an alternative approach to the problem of Vlasov plasma oscillations which is based on the Fourier transformed velocity space.…”