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Formation of ion-acoustic collisionless shocks was studied by numerically solving the one-dimensional Vlasov equation. The shock propagates at about the local ion-acoustic speed, and faster than the ion-acoustic speed ahead; the oscillations behind the shock show ion-acoustic dispersion. A precursor appears with a speed of 1.6 times the shock speed. Different temperature and density ratios produce qualitatively different flows, and they are analyzed in detail.
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Sakanaka et al. (1971) studied this question.
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