Nonlinear oscillations of rarified plasma

The paper basically is a review of a number of studies devoted to the theory of nonlinear motions of plasma under conditions where collisions between particles do not play a determining role.The problem is formulated in the introduction. It concerns the evolution in time of an initial perturbation of finite amplitude. The resulting physical picture will depend on these competing processes: nonlinear increase of wave steepness, dispersion, absorption and instability. In a number of cases where adsorption and instability are insignificant, it is possible to obtain an idea of the character of the nonlinear motions by applying the appropriate linear "dispersion law".The second Section presents certain specific types of nonstationary nonlinear motions permitting exact mathematical solution, namely: nonlinear oscillations of electrons at zero temperature, nonlinear motion of plasma across a strong magnetic field, and ion waves of finite amplitude in non-isothermal plasma where pi ≪ pe. In a number of instances, evolution of the initial perturbation leads to the formation of a multi-component current, some peculiarities of which are discussed in the third Section. In the next Section stationary nonlinear waves are described, i.e. waves not changing their form with time. In a particular case they are the so-called "solitary waves", similar to waves on the surface of heavy liquid in a channel of finite depth. The possibility of the existence of such waves requires linear laws of dispersion of a specific character. The possibility of "solitary" waves of rarefaction is pointed out.The question of absorption of waves in rarified plasma is then discussed. An approximate "quasi-linear" method is developed which permits the kinetic considerations in the absorption of waves of finite amplitude to be simplified. The method consists in the representation of the distribution function f(r, v, t) as a sum of rapidly and slowly varying terms. In the equation for the slowly varying term a quadratic average effect of fast oscillations is taken into consideration. This method is applied to two particular problems: the absorption of Langmuir electronic oscillations (in the limit of very small amplitude the equation for wave-damping goes over into the well-known formula for the so-called "Landau-damping"), and the cyclotronic absorption of transversely polarized waves which propagate along the constant magnetic field.In the last section some types of instabilities of non-linear motions are shown. In addition to the instabilities associated with the multi-component motion, it is shown that waves in a magnetic field (in particular, solitary waves) are unstable if their amplitude exceeds a certain critical value which decreases as the plasma temperature decreases.