Damping of transversal plasma-electron oscillations and waves in low-collision electron-ion plasmas
Previously developed method for finding asymptotic solutions of Vlasov equations using two-dimensional (in coordinate x and time t) Laplace transform is here applied to consider transversal oscillations and waves in low-collision quasi-neutral (n_i \simeq n_e) Maxwellian electron-ion plasmas. We obtain two branches of electron waves: the ubiquitous one of high-frequency and high-velocity oscillations and the unusual low-velocity one. Taking into account Coulomb collisions in the limit m_e << m_i, \bar{v_i} << \bar{v_e}, and T_e m_e << T_i m_i results in expressions for transversal plasma-electron oscillation/wave decrements with a damping of the low-velocity electron branch \sim n_i^{1/3}/\bar{v}_e^{4/3}, where n_i is the ion density and \bar{v}_e is the mean electron velocity. It ought to rehabilitate Vlasov principal value prescription for relevant integrals, but to supplement it with representation of an asymptotical solution as a sum of exponents (not a single one). "Non-damping" kinematical waves in low-collision plasma transform in the damping ones at reasonably chosen iteration process.
Solving non-linear equations of longitudinal and transverse electron waves in collisionless Maxwellian plasma
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.
A New Look at the Landau's Theory of Spreading and Damping of Waves in Collisionless Plasmas
The theory of plasma waves and Landau damping in Maxwellian plasmas, Landau's ``rule of pass around poles'' include doubtful statements, particularly related to an artificial ``constructing'' of the dispersion equation, what should allow the possibility of its solution otherwise not existing at all, and the possibility of analytical continuations of corresponding very specific ruptured functions in the one-dimensional Laplace transformation, used by Landau, what is the base of his theory. 
We represent, as an accessible variant, a more general alternative theory based on a two-dimensional Laplace transformation, leading to an asymptotical in time and space solution as a complicated superposition of coupled damping and {\em non-damping \/} plane waves and oscillations with different dispersion laws for every constituent mode. This theory naturally and very simply explains paradoxes of the phenomenon of plasma echo. We propose for discussion a new ideology of plasma waves (both electron and ion-acoustic waves) qualitatively different from the traditional theory of Landau damping for non-collisional as well as for low-collisional plasmas.