Two integrable classes of Emden-Fowler equations with applications in astrophysics and cosmology
Mancas, Stefan C
Rosu Barbus, Haret-Codratian
"We show that some Emden–Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d2z/dχ2=Aχ−λ−2zn for λ=(n−1)/2 (class 1), and λ=n+1 (class 2). 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 the form d2ν/dt2+adν/dt+b(ν−νn)=0, where the coefficients a, b depend only on λ,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 1 and 2, and one n = 5 case from class 1, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n = 2, the EF equations can be studied using the Painlevé reduction method, since they are a particular case of equations of the type d2z/dχ2=F(χ)z2 , where F(χ) is the Kustaanheimo-Qvist function."
PublisherWalter de Gruyter GmbH
Weierstrass Elliptic Function