Periodic, Solitary and Interacting Nonlinear Waves in Collisionless Plasmas

This dissertation is devoted to a study of so -called Bernstein-Greene-Kruskal (BGK) waves, a class of exact solutions of the nonlinear Vlasov-Poisson-Ampere equations for collisionless plasmas. Building upon previous work, we develop a simple but powerful formalism that facilitates methodical investigation of the types and properties of small amplitude BGK plasma waves that can exist nearby a given collisionless plasma equilibrium. We show that any physically relevant plasma equilibrium supports spatially -periodic BGK waves which are described by the Vlasov dispersion relation in the small amplitude limit. Furthermore, we show that such equilibria are characterized by a finite set of critical velocities v_sp{c} {(i)}, i = 1, 2... which are the velocities with which BGK solitary waves of vanishingly small amplitude can propagate in the plasma. The existence of these exact nonlinear waves illustrates the fundamental incompleteness of the linear Vlasov-Landau theory of plasma waves since, by virtue of particle trapping, they neither damp nor grow even when their amplitude is arbitrarily small. After describing this broad spectrum of small amplitude BGK waves, we then develop limited results concerning their interactions with one another. By studying the nonintegrable Hamiltonian system for a single charged particle in the field of two spatially-periodic electrostatic waves, we demonstrate that small amplitude spatially-periodic BGK waves of sufficiently different velocity satisfy a nonlinear superposition principle, in which the fields superimpose linearly while the self-consistent distribution functions combine according to a somewhat more complicated rule. This superposition principle explains, in part, recent large scale numerical calculations which suggest that superpositions of such waves are necessary for the proper description of the asymptotic states for the evolving two-stream and bump-on-tail instabilities, as well as for a large amplitude electrostatic wave that undergoes nonlinear Landau damping.