Generalized Thomas-Fermi Equations as the Lampariello Class of Emden-Fowler Equations

TL;DR

This paper studies a family of generalized Thomas-Fermi equations derived from Emden-Fowler equations, demonstrating their non-integrability and analyzing their dynamical systems to understand their behavior.

Contribution

It introduces a one-parameter family of equations linked to Lampariello's parameter, extending the standard Thomas-Fermi model and analyzing their integrability and dynamics.

Findings

The entire family is non-integrable due to Abel equation invariants.

Phase-plane analysis reveals similar dynamical behavior across the class.

Standard Thomas-Fermi equation is a special case with p=1.

Abstract

A one-parameter family of Emden-Fowler equations defined by Lampariello's parameter p which, upon using Thomas-Fermi boundary conditions, turns into a set of generalized Thomas-Fermi equations comprising the standard Thomas-Fermi equation for p=1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel equations whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of equations for the standard Thomas-Fermi equation and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.