Integral equations for disturbance propagation in linearized Vlasov plasmas Numerical results

Abstract

Space-time responses of linearized Vlasov plasmas on the basis of multiple integral equations are considered. An initial-value problem for Vlasov-Poisson/Ampere equations is reduced to one integral equation and the solution is expressed in terms of a forcing function and its space-time convolution with the resolvent kernel. The forcing function is responsible for the initial disturbance and the resolvent is responsible for the equilibrium velocity distribution. For Maxwellian equilibrium distribution, a closed-form solution of the resolvent kernel equation is still unknown but the equation is eligible for computer calculations. Three types of exact analytical solutions of the space-time resolvent equations are shown to relate them to Maxwellian plasmas. Numerical calculations reveal the nature of the plasma response as a compound of a diffusive transition, being essentially a plasma oscillation mode with plasma frequency, a Gaussian type of amplitude profiles, and also a damped dispersive wave mode. The plasma response appears immediately in the whole space of x and zeros (nodes) travel according to the diffusion law, at least for long times. By use of the resolvent equations, time-reversibility and space-reflexivity can be revealed. The step-density disturbance of electron Maxwellian plasmas appears to be the electric current forcing function, which is proportional to the Maxwellian plasma kernel; hence the resolvent is the plasma response to the step-density disturbance. From inspections of the series representations of Maxwellian resolvent and its Fourier transform, a symmetry property with respect to the transformation is found. It is used for constructing approximate formulae for the resolvent kernels.