Langmuir oscillations in a cold inhomogeneous plasma

One-dimensional nonlinear Langmuir oscillations in a cold inhomogeneous plasma are considered. The spatial distribution of the amplitude of oscillations is set by the initial disturbance and can be arbitrary, i.e., it is not in any way connected with the inhomogeneity of the plasma. Taking place at small spatial gradients of the initial disturbance because of a growing with time phase shift of oscillations of different electrons, the effect of formation of narrow density peaks is illustrated. The density peaks always move in the direction of a decrease of ion concentration. In this movement their amplitude can grow or decrease depending on the direction of the initial disturbance gradient. A number of electrons, forming density peaks, reduce with a diminution of plasma inhomogeneity but the peak amplitudes always increase unlimitedly with time up to the self-intersection of electron trajectories. An equation, making it possible to determine the instant of trajectory intersection in case of small plasma inhomogeneity of arbitrary kind and arbitrary gradients of the initial disturbance, is obtained. It is shown by a numerical calculation that the analytical relations obtained are not essentially changed if one takes into account the nonlinearity of corresponding differential equations. copyright 1996 The American Physical Society