Bifurcation of BGK waves in a plasma of cold ions and electrons

For the simple model of cold electrons streaming against cold ions the complete set of nonlinear stationary waves is expressed in terms of elliptic functions. The conditions for their dynamical connection to a uniform neutral plasma state are taken into account, and the conditions for the neglect of the magnetic field are analysed. The range of existence of stationary waves is found to be confined to the stable regime of the two-stream instability, but covers only part of it. All nonlinear BGK waves that are found within the limits of the model can be shown to bifurcate from the two-stream instability, some of them also exhibiting secondary and further bifurcations. As an exceptional case, all bifurcations can be treated exactly. Close to the linear regime, all nonlinear modes turn out to be unstable. The corresponding instability is caused by a wave decay that transports energy from low to high wavenumbers of the Fourier modes constituting the wave. From the two-stream solutions four-stream solutions with exactly vanishing magnetic field are derived. (author)