Zur nichtlinearen Theorie der Strahl-Plasma-Wechselwirkung III. Vorbereitung der numerischen Rechnungen

In order to prepare the system of ordinary differential equations, describing approximately the non-linear behaviour of a beam-plasma system, for numerical solution, at first the dispersion relation is analized. In particular it is shown, that in the case of a three component plasma the complex wave number k = k_r + i_γ can be found without difficulties as a function of frequency ω and the plasma parameters, if the electron densities n_b and n_p are treated as the two dependent quantities. The expressions on the right hand sides of the differential equations are transformed in such a way, that all complex integrals can be found as functions of one basic integral, tabulated in the literature, not only for negative values of γ (amplification) but also for positive values of γ (evanescence). In this way it will be possible, to describe the behavior of the wave amplitude not only for short distances from the input, where amplification of the wave takes place, but also for larger distances, where nonlinear Landau-damping may occur.