The 1D Vlasov-Poisson System Driven by an Externally Applied Time-dependent Electric Field

The two commonly recognized standard methods of solving the 1D Vlasov-Poisson system, the Jackson and van Kampen-Case methods, have not been formulated to account for an externally applied time-dependent electric field. The Jackson method assumes the integrand decays in the complex lower half-plane at infinity and the contour portion of the Bromwich integral is discarded while the poles are encircled and evaluated as residues. If an externally applied electric field is included in the Jackson method, the Laplace transform of the externally applied field can produce divergences in the lower-half plane at infinity and the method breaks down. It is unclear how the inclusion of an externally applied electric field affects the eigenvalue problem of the van Kampen-Case method, but even if it is incorporated its time evolution would not be immediately clear as the solution would be left as an opaque integral due to the van Kampen continuum. On the other hand, the Cauchy-type integral method recently introduced by the authors is well suited to incorporate an externally applied electric field as the divergences in the lower-half plane are properly accounted for in the Cauchy splitting procedure and the time evolution is explicitly found. We show examples of externally applied electric fields in the form of a Heaviside function, a Gaussian, and a sinusoid, and propose a method for determining the Landau frequency and damping rate for a plasma at equilibrium using an external field and the corresponding plasma response.