Lie symmetry analysis of the Grad-Shafranov equation

TL;DR

This paper applies Lie symmetry analysis to the Grad-Shafranov equation in plasma physics, classifying its symmetries and providing a method to generate new solutions.

Contribution

It offers a symmetry classification of the Grad-Shafranov equation with arbitrary functions and constructs an optimal system of subalgebras for solution generation.

Findings

Symmetry classification of the Grad-Shafranov equation with arbitrary functions.

Development of an optimal system of one-dimensional subalgebras.

Framework for constructing new solutions using symmetry methods.

Abstract

The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose the Grad-Shafranov equation which may illustrate the reciprocal advantage of this interaction between plasma physics and symmetry techniques. A symmetry classification of the Grad-Shafranov equation with two arbitrary functions $F(u)$ and $G(u)$ of the unknown variable $u=u(x,t)$ is given. The optimal system of one-dimensional subalgebras is performed. This latter provides a process for building new solutions for the equation.