Parametric instability in the region of low-frequency hybrid resonances

The non-linear longitudinal oscillations of a homogeneous plasma embedded in a weakly inhomogeneous magnetic field in the presence of an external high frequency field are investigated. The non-linearity of oscillations is manifested in two ways: (a) allowance for the terms (∇) in the equations of motion and (b) dependence of the plasma natural oscillation frequency on the displacement amplitude (or, otherwise, the loss of isochronity). It is demonstrated that the terms (∇) do not restrict the amplitude increase in the regions of hybrid resonances. The non-linear terms arising in the equations of motion governed by the inhomogeneity of 0, item (a), restrict the oscillation amplitude in the resonant areas and, item (b), lead to a parametric build-up of oscillations. The conditions for the parametric instability and the growth rates of unstable oscillations are determined.