Potential symmetry and invariant solutions of Fokker-Planck equation in cylindrical coordinates related to magnetic field diffusion in magnetohydrodynamics including the Hall current

TL;DR

This paper applies Lie group symmetries to derive invariant solutions of a Fokker-Planck equation in cylindrical coordinates, modeling magnetic field diffusion in magnetohydrodynamics with Hall current, revealing new exact solutions.

Contribution

It introduces a novel symmetry-based method to find solutions of magnetohydrodynamic equations including Hall current effects in cylindrical geometry.

Findings

Derived new exact solutions for the Fokker-Planck equation

Obtained invariant solutions considering time-dependent flow and Hall current

Identified potential symmetries leading to simplified equations

Abstract

Lie groups involving potential symmetries are applied in connection with the system of magnetohydrodynamic equations for incompressible matter with Ohm's law for finite resistivity and Hall current in cylindrical geometry. Some simplifications allow to obtain a Fokker-Planck type equation. Invariant solutions are obtained involving the effects of time-dependent flow and the Hall-current. Some interesting side results of this approach are new exact solutions that do not seem to have been reported in the literature.