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We have considered an expansion of solutions of the non-linear equations forboth longitudinal and transverse waves in collisionless Maxwellian plasma inseries of non-damping overtones of the field E(x,t) and electron velocitydistribution function f=f(0) +f(1) where f(0) is background Maxwellian electrondistribution function and f(1) is perturbation. The electrical field andperturbation f(1) are presented as a series of non-damping harmonics withincreasing frequencies of the order n and the same propagation speed. It isshown presence of recurrent relations for arising overtones. Convergence of theseries is provided by a power law parameter series convergence. There areproposed also successive procedures of cutting off the distribution functionf(1) to the condition of positivity f near the singularity points where kineticequation becomes inapplicable. In this case, at poles absence the solutionreduces to non-damping Vlasov waves (oscillations). In the case of transversewaves, dispersion equation has two roots, corresponding to the branches of fastelectromagnetic and slow electron waves. There is noted a possibility ofexperimental testing appearing exotic results with detecting frequencies andamplitudes of n-order overtones.

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