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

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.