NONLINEAR OSCILLATIONS IN A VLASOV-MAXWELL PLASMA

In the absence of spatial variations, the Ylasov-Maxwell equations are solved exactly allowing for an initial plasma current to generate large-amplitude, oscillatory, electric fields in the presence of a uniform external magnetic field. These results are extended to include the effects of weak spatial inhomogeneities for the case in which no external magnetic field is present. The analysis is carried out to first order in the small parameter, λ ∼(Debye length/length scale of spatial variations), by means of a multiple-time, multiple-space, perturbation scheme. Due to the presence of weak spatial variations, a magnetic field is generated oscillating at the plasma frequency, whereas the zero-order electric field is modified by oscillatory behavior at the second harmonic of the plasma frequency. The distribution function is obtained to order λ in situations where the zero-order electric field is initially curl free; under these circumstances no first-order magnetic field is generated and the problem remains electrostatic to order λ. In addition, a nonlinear differential equation is derived describing the long time (t ∼ 1/λ) behavior of the zero-order distribution function; the linear stability of this equation is discussed.