Nonlinear oscillations in a cold plasma

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Abstract
In a Maxwell-fluid description large-amplitude electrostatic and electromagnetic oscillations in a cold plasma are analysed in situations where the spatial variations are one-dimensional and the ions form a fixed neutralizing background. In the electrostatic approximation a Lagrangian description following the electron motion is adopted, and exact solutions obtained in these variables in situations where multistream flow does not develop. The inversion to Eulerian coordinates is carried out explicitly for the particular example of an initial sinusoidal perturbation in density. A model describing the dispersive modifications of (weak) thermal effects is analysed showing the phase-mixing of (moderately) large-amplitude initial disturbances. After sufficient time, the electron fluid becomes stationary and a low level of electric field remains balancing the force due to pressure variations. In addition, for the case of nonlinear electrostatic oscillations perpendicular to a static magnetic field, the exact solution in Lagrangian variables shows that for a cold plasma coherent oscillations at the upper hybrid frequency are maintained indefinitely over the regions of initial excitation. Electromagnetic oscillations are considered as developing from small initial values on the background of the large-amplitude longitudinal oscillations already calculated. The resulting wave equation is analysed for several limiting cases. It is shown that for sufficiently short wavelengths of the transverse fields there exists an infinite number of narrow ranges of the wave number in which the transverse oscillations are unstable. The growth is ultimately limited by the magnetic force which was neglected in the description of the longitudinal motion. Finally, stationary solutions are studied. In the electrostatic approximation a very special class of periodic Bernstein-Greene-Kruskal waves without trapped particles is obtained. Again the electromagnetic solutions are unstable (in space), the growth being limited by the same effect as in the time-dependent problem. A simple class of special stationary solutions is obtained for the complete problem including all magnetic force terms.

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