Nonlinear electrostatic waves in collisionless plasmas.

Abstract

We report results concerning small amplitude Bernstein-Greene-Kruskal (BGK) waves, which are exact undamped traveling wave solutions of the nonlinear Vlasov-Poisson-Amp\`ere equations for collisionless plasmas. Building upon previous work, we first develop a simple but powerful formalism that facilitates a methodical investigation of the types and properties of small amplitude BGK plasma waves that can exist near a given collisionless plasma equilibrium. Using this formalism, we then show that any physically relevant spatially uniform plasma equilibrium supports nonlinear spatially periodic BGK waves that are described by the Vlasov dispersion relation in the small amplitude limit. We demonstrate also that these equilibria are characterized by a discrete set of critical velocities ${\mathit{v}}_{\mathit{c}}^{(\mathit{i})}$, i=1,2,..., at which BGK solitary waves of vanishingly small amplitude can propagate in the plasma. The existence of these exact nonlinear spatially periodic and solitary wave solutions illustrates the fundamental incompleteness of the linear Vlasov-Landau theory of plasma waves since, by virtue of particle trapping, these nonlinear waves neither damp nor grow even when their amplitude is arbitrarily small.