Solitary, Double-Layer and Periodic Waves of Arbitrary Amplitude in Nonlinear Collisionless Plasmas

This dissertation presents an investigation of traveling wave solutions of the nonlinear Vlasov-Poisson -Ampere equations, which govern longitudinal particle motion along magnetic field lines in a collisionless plasma. The new work builds upon previous results that use the so-called mechanical potential formalism to derive small amplitude spatially periodic, solitary, and double-layer wave solutions of the Bernstein-Greene-Kruskal (BGK) type. Here the mechanical potential formalism is extended to develop a general framework for finding BGK solutions that allows for the incorporation of specific information about plasma particle trajectories in the presence of the wave. This framework is then applied to the physically relevant problem of a charged particle beam in interaction with a traveling solitary wave. It is shown that the problem may be reduced to a two-dimensional set of nonlinear algebraic equations, which are then solved via a rigorous bifurcation analysis using the Liapunov-Schmidt dimensional reduction technique. The results yield a branch of exact small amplitude asymmetric solitary wave solutions that bifurcates from the manifold of Vlasov plasma equilibria, parameterized both by the velocity and density of the beam. After describing the asymmetric solitary waves, we then demonstrate the existence of branches of large amplitude solitary, double -layer, and periodic waves by applying the mechanical potential formalism in conjunction with simple numerical techniques. Evidence is given to show that these branches of large amplitude waves are merely the continuation of the branches of small amplitude waves that have been shown previously to bifurcate from the manifold of Vlasov equilibria. Once these solutions have been established, the effects of the smoothness properties of the self-consistent particle distribution functions on the spectrum of traveling wave solutions is probed. Results are given to show that branches of solitary, double-layer, and periodic waves of both small and large amplitude can exist in plasmas described by distribution functions that are quite non-smooth. Motivation for the new work presented herein is given through a summary of satellite observations of solitary waves in the earth's magnetosphere, and the importance of the new solutions to our understanding of the mechanism for auroral particle acceleration is discussed.