Solution of the first order quasi-linear equation that defines the evolution of plasma turbulence. PMM vol. 40, n{ring equal to} 5, 1976, pp. 823-833

An asymptotic solution of the Cauchy problem is obtained for the first order quasi-linear equation. The field of characteristic curves is constructed. It is shown that for fairly considerable times the solution is discontinuous, but tends to a smooth stationary distribution. Numerical calculations obtained by the method of characteristics are presented. Results of the asymptotic and numerical analysis are in good agreement. A homogeneous boundless plasma subjected to a strong high-frequency radiation [1] or a beam of particles [2] is unstable. Exponentially increasing with time t electromagnetic oscillations with a wave vector k from some phase space region are induced in it during the initial linear stage. Nonlinear processes become substantial later when the energy spectral density w (k, t) reaches a high level. Such processes result in the saturation of the electromagnetic wave energy. The state of plasma under such conditions is called turbulent [3-5]. Investigation of the evolution of the energy spectral density of turbulent noise serves as the base for determining the form of functions of plasma particle distribution and the law governing the absorption and dispersion by the plasma of beams of particles or of powerful radiation. These questions are fundamental in the problem of the utilization of powerful light beams and of relativistic electrons for heating plasma.