Two integrable classes of Emden-Fowler equations with applications in astrophysics and cosmology

TL;DR

This paper classifies certain Emden-Fowler equations relevant to astrophysics and cosmology into two integrable classes, finds their invariants, reduces them to simpler forms, and provides explicit solutions including elliptic functions.

Contribution

It identifies two new integrable classes of Emden-Fowler equations, derives their invariants, and presents closed-form parametric solutions with applications in astrophysics and relativity.

Findings

Two integrable classes of EF equations are characterized.

Explicit parametric solutions including elliptic functions are provided.

EF equations with n=2 can be analyzed via Painlevé reduction.

Abstract

We show that some Emden-Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d^2z/d{\chi}^2=A \chi^{-\lambda-2}z^n for \lambda=(n-1)/2 (class one), and \lambda=n+1 (class two). We find their corresponding invariants which reduce them to first order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator form d^2\nu/dt^2 +ad \nu/dt +b(\nu-\nu^n)=0, where the coefficients a,b depend only on \lambda, n. For both classes, we write closed-form solutions in parametric form. The illustrative examples from astrophysics and general relativity correspond to two n=2 cases from class one and two, and one n=5 case from class one, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n=2, EF equations can be studied using the Painlev\'e reduction method, since they are a particular case of equations of the type d^2z/d{\chi}^2=F(\chi)z^2, where F(\chi) is the Kustaanheimo-Qvist function