Notes on the exact solutions and singularities of an X-point collapse in Hall magnetohydrodynamics

TL;DR

This paper corrects a previous analysis of dissipationless Hall magnetohydrodynamics, providing a complete solution using Jacobi elliptic functions and clarifying the energy considerations involved.

Contribution

It offers the correct energy analysis and a full solution of Hall MHD equations, improving upon prior partial solutions that used Weierstrass elliptic functions.

Findings

Corrected the energy analysis in Hall MHD solutions.

Provided a complete solution using Jacobi elliptic functions.

Clarified the mathematical structure of X-point collapse in Hall MHD.

Abstract

A recent paper by A.~Z.~Janda [J.~Math.~Phys.~{\bf 59}, 061509 (2018)] presented a partial solution of the equations of dissipationless Hall magnetohydrodynamics (MHD) in terms of the Weierstrass elliptic function. Unfortunately, an error crept up in the analysis, where the kinetic and potential energies of the initial Hall MHD state were misidentified. The present comment presents the correct energy analysis and offers a complete solution of dissipationless Hall magnetohydrodynamics in terms of the Jacobi elliptic functions.