Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

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Abstract
Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

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