Similarity Inner Solutions for the Pulsar Equation

TL;DR

This paper uses Lie symmetries and singularity analysis to classify and solve the Pulsar equation near neutron star surfaces, providing analytic inner solutions.

Contribution

It introduces a symmetry-based classification of the Pulsar equation and derives explicit inner solutions using Lie invariants and the ARS algorithm.

Findings

Identified six different Lie algebras for the Pulsar equation.

Derived analytic inner solutions as Laurent expansions.

Demonstrated reduction of the PDE to ODEs using symmetries.

Abstract

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.