Time-asymptotic traveling-wave solutions to the nonlinear Vlasov–Poisson–Ampère equations

We consider the Vlasov–Poisson–Ampère system of equations, and we seek solutions for the electric field E(x ,t) that are periodic in space and asymptotically almost periodic in time. Introducing the representation E(x ,t)5T(x ,t)1A(x ,t) ~where T and A are, respectively, the transient and time-asymptotic parts of E! enables us to decompose the nonlinear Poisson equation into a transient equation and a time-asymptotic equation. We then study the latter in isolation as a bifurcation problem for A with the initial condition and T as parameters. We show that the Fréchet derivative at a generic bifurcation point has a nontrivial null space determined by the roots of a Vlasov dispersion relation. Hence, the bifurcation analysis leads to a general solution for A given ~at leading order! by a discrete superposition of traveling-wave modes, whose frequencies and wave numbers satisfy the Vlasov dispersion relation, and whose amplitudes satisfy a system of nonlinear algebraic equations. In applications, there is usually a finite number of roots to the dispersion relation, and the equations for the time-asymptotic wave amplitudes reduce to a finite dimensional bifurcation problem in terms of the amplitude of the initial condition. © 1999 American Institute of Physics. @S0022-2488~99!01208-6#