The emission of plasma waves by beams of electrons travelling in a plasma is a phenomenon of critical importance in applied plasma physics (for instance in problems directly related to the achievement of controlled nuclear fusion) and also astrophysical research (e.g. in the theory of solar radio bursts). In principle, the mechanisms involved are all contained in the Boltzmann-Vlasov equation, where the field is the self-consistent electromagnetic field produced by the interaction between beam and plasma. Unfortunately this celebrated equation cannot be solved directly, because both the analytical and numerical methods that can deal with this equation are plagued by secular terms which restrict the time domain of validity of the solutions to a few thousand plasma periods. In all applications of interest this domain is far too small; indeed in all astrophysical cases it is quite negligible compared with the duration of the observed phenomena (it is even much shorter than the time resolution of present-day equipment, such as dynamic spectrographs).
Fine structure of type III solar radio bursts from Langmuir wave motion in turbulent plasma