Bernstein–Greene–Kruskal (BGK) equilibria for a Vlasov plasma consisting of a periodic structure exhibiting depressions or ‘‘holes’’ in phase space are under consideration. Marginal stability analysis indicates that such structures are unstable when the system contains at least two holes. An Eulerian numerical code is developed allowing noiseless information on the long time phase space behavior (about 103ω−1p) to be obtained. Starting with equilibria with up to six holes, it is shown that the final state is given by a structure with only one large hole, the initial instability inducing coalescences of the different holes. On the other hand, starting with a homogeneous two-stream plasma it is shown that, in a first step, a BGK periodic structure appears with a number of holes proportional to the length of the system, followed, in a second step, by a coalescence of the holes to always end up with the above mentioned one large hole structure.
The Schrödinger equation describes the motion of a particle in a statistical sense. It consequently possesses the two main properties of the Vlasov equation (dynamic and statistic) and can replace this last equation provided we take sophisticated initial conditions. The scheme must be considered as a new attempt to discretize intelligently the amount of information contained in the phase space distribution and to stop, without destroying it, the flow of information which usually goes to high wavenumbers in velocity space. The method is applied to the breaking of highly nonlinear waves in a cold plasma (usually treated by the Lagrangian method) and to double beam instability. It is shown that such an Eulerian scheme works quite well with a much smaller number of discretized functions than are required in the regular Fourier—Fourier or Fourier-Hermite methods. The central point is the introduction of the phase space Wigner distribution function which is a useful mathematical tool in spite of its poor physical properties.
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