Abstract
Abstract We show that some Emden–Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d 2 z / d χ 2 = A χ − λ − 2 z n ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=A{\chi^{-\lambda-2}}{z^{n}}$ for λ = ( n − 1 ) / 2 $\lambda=(n-1)/2$ (class 1), and λ = n + 1 $\lambda=n+1$ (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 d 2 ν / d t 2 + a d ν / d t + b ( ν − ν n ) = 0 ${\mathrm{d}^{2}}\nu/\mathrm{d}{t^{2}}+a\mathrm{d}\nu/\mathrm{d}t+b(\nu-{\nu^{n}})=0$ , where the coefficients a , b $a,b$ depend only on λ , n $\lambda,n$ . 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 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 d 2 z / d χ 2 = F ( χ ) z 2 ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=F(\chi){z^{2}}$ , where F ( χ ) $F(\chi)$ is the Kustaanheimo-Qvist function.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have