Particular solutions of the inverse problem for 1D Vlasov-Maxwell equilibria using Hermite polynomials

TL;DR

This paper develops a method using Hermite polynomials to solve an inverse problem for 1D Vlasov-Maxwell equilibria, providing convergent series solutions and insights into the non-negativity of the distribution function.

Contribution

It introduces a Hermite polynomial-based approach to invert Weierstrass transforms in plasma equilibrium problems, with general applicability beyond the specific magnetic field considered.

Findings

Hermite polynomial series converge for the inverse problem

Method applicable to various smooth pressure functions

Discussion on conditions for distribution function non-negativity

Abstract

We present the solution to an inverse problem arising in the context of finding a distribution function for a specific collisionless plasma equilibrium. The inverse problem involves the solution of two integral equations, each having the form of a Weierstrass transform. We prove that inverting the Weierstrass transform using Hermite polynomials leads to convergent infinite series. We also comment on the non-negativity of the distribution function, with more detail on this in Allanson $\textit{et al., Journal of Plasma Physics}$, vol. 82 (03), 2016. Whilst applied to a specific magnetic field, the inversion techniques used in this paper (as well as the derived convergence criteria and discussion of non-negativity) are of a general nature, and are applicable to other smooth pressure functions.