Higher order approximations in the theory of longitudinal plasma oscillations

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Abstract
The non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma is considered for the case in which a single wave of small, but finite amplitude is excited initially. The Vlasov–Poisson equations are solved using the method of strained co-ordinates in which the independent variable t, the electric field and the distribution function are expanded in the form of asymptotic series, the terms of which are founded by an iterative procedure. An ordering parameter e is introduced, which is proportional to the initial amplitude of the electric field given by linear theory. Differential equations are derived which can be solved sequentially to obtain uniformly valid solutions to all orders in ε. Solutions are given to second order and applied to the case in which the background distribution function is Maxwellian. It is found that the changes in the real and imaginary part of the frequency are small in comparison to the values obtained in the linear theory; that the free-streaming terms decay exponentially in time with a damping rate proportional to ε2, in contrast with the linear theory where they are Un- damped; and that the analysis allows us to calculate the changes in the background distribution function for large time, resulting from particle-wave interactions.

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