# Mathematical analysis of equations in plasma physics

## Abstract

In this paper, two equations from plasma physics are analyzed using two different mathematical procedures to yield information of interest for fusion energy. In the first case, Lie’s technique of computing symmetries of differential equations is applied to a specific case of the Grad-Shafranov equation. The case considered contains the majority of exact solutions from the literature. The full symmetry group is computed and new group-invariant solutions are obtained from these symmetries. The basic results and methods behind this technique are given along with several plots of the level sets or flux surfaces of the new solutions. In addition, a mathematical technique which was first used to prove the non-existence of solitons in quantum field theory is employed to derive an integral relation for any solution of the Sinh-Poisson equation. The original technique is modified to allow for a finite boundary and results are computed for two different boundary geometries.