General dispersion equation for oscillations and waves in non-collisional Maxwellian plasmas

Abstract

We propose a new and effective method to find plasma oscillatory and wave modes. It implies searching a pair of poles of two-dimensional (in coordinate $x$ and time $t$) Laplace transform of self-consistent plasma electric field $E(x,t) \to E_{p_1p_2}$, where $p_1 \equiv -i \omega$, $p_2 \equiv i k$ are Laplace transform parameters, that is determining a pair of zeros of the following equation $$\frac1{E_{p_1p_2}} = 0 .$$ This kind of conditional equation for searching double poles of $E_{p_1p_2}$ we call ``general dispersion equation'', so far as it is used to find the pair values ($\omega^{(n)}, k^{(n)}$), $n=1, 2, ...$ . It differs basically from the classic dispersion equation $\epsilon_l(\omega,k) = 0$ (and is not its generalization), where $\epsilon_l$ is longitudinal dielectric susceptibility, its analytical formula being derived according to Landau analytical continuation. In distinction to $\epsilon_l$, which is completely plasma characteristic, the function $E_{p_1p_2}$ is defined by initial and boundary conditions and allows one to find all the variety of asymptotical plasma modes for each concrete plasma problem. In this paper we demonstrate some possibilities of applying this method to the simplest cases of collisionless ion-electron plasma and to electron plasma with collisions described by a collision-relaxation term $-\nu f^{(1)}$.