Abstract Details
Abstracts
Author: Frank M. Lee
Requested Type: Consider for Invited
Submitted: 2023-03-24 15:07:09
Co-authors: B.A. Shadwick
Contact Info:
University of Nebraska-Lincoln
855 North 16th Street
Lincoln, NE 68588
USA
Abstract Text:
We describe a new method for solving the linearized 1D Vlasov-Poisson equation that remedies critical flaws of the two standard methods. The Landau approach [1] involves deforming the Laplace inversion contour around the poles closest to the real axis due to the analytically-continued dielectric function in order to determine the long-time behavior. Jackson generalized this [2] to encircle all poles while sending the contour to infinity, assuming its contribution vanishes in the limit, which is not true in general. This gives incorrect solutions for physically reasonable configurations; an error widely reproduced in standard textbooks. We show examples clearly revealing the error, simultaneously obtaining previously unseen time-evolutions. Van Kampen's method [3] is a stationary wave approach that leaves the solution as an integral representing a continuum of waves. Case generalized this [4] to include unstable systems and predicts a decaying discrete mode accompanying each unstable mode. We show these decaying modes are not present in the solution due to an unconditional cancellation with part of the continuum. The key to our new method is the careful complex analysis of Cauchy-type integrals as described by Gakhov [5] that are inseparable from the problem. This analysis can be readily applied in higher-dimensional, electromagnetic systems and represents a significant advance in fundamental plasma physics. This work provides a new technique for evaluating certain inverse Laplace transforms, thus having wide application in mathematical physics.
[1] L. D. Landau, On the vibration of the electronic plasma, J. Phys. USSR 10, 25 (1946)
[2] J. D. Jackson, Longitudinal plasma oscillations, J. Nucl. Energy C 1, 171 (1960)
[3] N. G. van Kampen, On the theory of stationary waves in plasmas, Physica 21, 949 (1955)
[4] K. M. Case, Plasma oscillations, Ann. Phys. 7, 349 (1959)
[5] F. D. Gakhov, Boundary Value Problems (Dover, New York, 1990)
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