Applications of Symmetry Methods to the Theory of Plasma Physics

Abstract

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 three different examples which may illustrate the reciprocal advantage of this "interaction" between plasma physics and symmetry techniques. The examples include, in particular, the complete symmetry analysis of system of two PDE's, with the determination of some conditional and partial symmetries, the construction of group-invariant solutions, and the symmetry classification of a nonlinear PDE.

Magnetohydrodynamic equilibria with incompressible flows: Symmetry approach

“…In Ref. 11, an analysis of the symmetry properties of a class of partial differential equations of relevance for plasma physics, including the GS equation as a special case, has been performed. In particular, in Ref.…”

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

“…Cicogna & Pegoraro (2015) derived a series of solutions to the GGS equation that describes the two-dimensional azimuthal symmetric plasma equilibrium in the presence of incompressible poloidal and toroidal plasma flows. For the Euler equation with a swirl, the symmetry properties of a particular GS equation describing the stationary, azimuthally symmetric magneto-static plasma equilibria were developed in Frewer et al (2007), White & Hazeltine (2009), Cicogna, Ceccherini & Pegoraro (2006), Cicogna, Pegoraro & Ceccherini (2010) and Cicogna (2008Cicogna ( , 2012. In Cicogna et al (2006), an analysis of the symmetry properties of a class of partial differential equations related to the GS equation was studied.…”

“…For the Euler equation with a swirl, the symmetry properties of a particular GS equation describing the stationary, azimuthally symmetric magneto-static plasma equilibria were developed in Frewer et al (2007), White & Hazeltine (2009), Cicogna, Ceccherini & Pegoraro (2006), Cicogna, Pegoraro & Ceccherini (2010) and Cicogna (2008Cicogna ( , 2012. In Cicogna et al (2006), an analysis of the symmetry properties of a class of partial differential equations related to the GS equation was studied. In Cicogna et al (2010), the conditional symmetries of the GS equation were introduced and some solutions to the GS equation describing the D-shaped toroidal plasma equilibria were constructed.…”

Symmetry classification of quasi-linear PDE’s containing arbitrary functions

“…[24], but without any details in the calculations and without any reference to the procedure, which is instead one of the main purposes of the present paper. In the same reference [24] one can also find some physical comment on the symmetry properties of the above equation.…”