A Novel Method for Solving the Linearized 1D Vlasov–Poisson Equation Using Cauchy-type Integrals

We describe a new method for solving the linearized 1D Vlasov-Poisson equation by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods and reveals previously unrecognized time evolutions. The Landau approximation involves deforming the Laplace inversion contour around the poles closest to the real axis due to the analytically-continued dielectric function to find the long-time behavior for a stable system, which is known as Landau damping. Jackson, in an attempt to fully solve the initial value problem, generalized this 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 and in some cases of infinite sums does not result in an asymptotic form; an error widely reproduced in standard textbooks. We show examples clearly revealing the error, simultaneously obtaining time evolutions that are not simply exponentials with arguments linear in time. The van Kampen method, which expresses the solution to the initial value problem for a stable system as a continuous superposition of waves, results in an opaque integral. Case generalized this to include unstable systems and predicts a decaying discrete mode for each growing discrete mode, resulting in an apparent contradiction to both the Jackson solution and ours. We show the decaying modes are not present in the time evolution due to an unconditional cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on an analytic continuation of the dielectric function, and naturally results in a correct asymptotic form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.