Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation

TL;DR

This paper analyzes the symmetry properties of the 1D Hall MHD system in cold plasma, deriving similarity solutions and exploring oscillating behaviors to deepen understanding of plasma dynamics.

Contribution

It identifies the Lie symmetries of the 1D HMHD system and constructs similarity solutions, providing new analytical insights into plasma behavior.

Findings

The system is invariant under a seven-dimensional Lie algebra.

Derived similarity transformations facilitate solution construction.

Presented various oscillating solutions for the HMHD system.

Abstract

We study the Lie point symmetries and the similarity transformations for the partial differential equations of the nonlinear one-dimensional magnetohydrodynamic system with the Hall term known as HMHD system. For this 1+1 system of partial differential equations we find that is invariant under the action of a seventh dimensional Lie algebra. Furthermore, the one-dimensional optimal system is derived while the Lie invariants are applied for the derivation of similarity transformations. We present different kinds of oscillating solutions.