Abstract
The problem of the stability of a one-dimensional inhomogeneous electron plasma in a fixed ion background is considered, using the water-bag model with N contours with only those instabilities appearing through the point ω = 0. For marginal stability, the system of N water-bag equations is shown to degenerate into a single equation, and the stability property is simply connected to the sign of the eigenvalues of the Schrödinger operator {k2(x) − d2/dx2), where ( − k2)1/2 is the local maximum wavenumber of instability. Finally, this criterion is checked for a simplified inhomogeneous two-stream situation described by a water-bag model with only two contours.
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