Solving non-linear equations of longitudinal and transverse electron waves in collisionless Maxwellian plasma

TL;DR

This paper develops a series expansion method to solve non-linear equations for electron waves in collisionless Maxwellian plasma, revealing overtones and conditions for non-damping solutions, with potential for experimental validation.

Contribution

It introduces a novel series expansion approach for non-linear plasma wave equations, including recurrent relations and convergence criteria, and discusses conditions for non-damping and experimental detection.

Findings

Series of non-damping overtones derived for plasma waves

Dispersion relation reveals two roots for transverse waves

Conditions for positivity and non-damping solutions established

Abstract

We have considered an expansion of solutions of the non-linear equations for both longitudinal and transverse waves in collisionless Maxwellian plasma in series of non-damping overtones of the field E(x,t) and electron velocity distribution function f=f(0) +f(1) where f(0) is background Maxwellian electron distribution function and f(1) is perturbation. The electrical field and perturbation f(1) are presented as a series of non-damping harmonics with increasing frequencies of the order n and the same propagation speed. It is shown presence of recurrent relations for arising overtones. Convergence of the series is provided by a power law parameter series convergence. There are proposed also successive procedures of cutting off the distribution function f(1) to the condition of positivity f near the singularity points where kinetic equation becomes inapplicable. In this case, at poles absence the solution reduces to non-damping Vlasov waves (oscillations). In the case of transverse waves, dispersion equation has two roots, corresponding to the branches of fast electromagnetic and slow electron waves. There is noted a possibility of experimental testing appearing exotic results with detecting frequencies and amplitudes of n-order overtones.