Integrability of the Tolman-Oppenheimer-Volkoff equation
type: Item
instance of: scholarly article
title: Integrability of the Tolman-Oppenheimer-Volkoff equation
author: Najmeh Khajoei
author: Claudia Valls
author: Jaume Giné
published in: Nonlinear Analysis. Real World Applications
publication date: 2022-11-11T00:00:00Z
review text: The paper studies a nonlinear model for a static star in general relativity from the view of the point of integrability. The considered model can be reduced to a planar system \[ \frac{d x}{d t}=y-x, \quad \frac{d y}{d t}=\frac{y}{1-2 x}\left(2-\frac{(5 \gamma-4) x}{\gamma-1}-\gamma y\right). \] After time scaling, it becomes a polynomial differential system \[ x^{\prime}=(y-x)(1-2 x), \quad y^{\prime}=y\left(2-\frac{5 \gamma-4}{\gamma-1} x-\gamma y\right).\tag{1} \] There are three main results. Firstly, they prove that system (1) with \(\gamma\neq\{0,1 \}\) has no global analytic first integrals, by analyzing the resonance relation of eigenvalues at the equilibrium. Secondly, they show that system (1) has a a Liouville first integral if and only if \(\gamma\in \{0,2/3,4/5,5/6\}\). To this end, the authors transform system (1) into a quadratic Lotka-Volterra system and apply the classification of Liouville first integrals for Lotka-Volterra system in [\textit{J. Moulin Ollagnier}, Qual. Theory Dyn. Syst. 2, No. 2, 307--358 (2002; Zbl 1084.34001)]. Finally, they further prove that it has no Puiseux first integrals with the help of a recent work in [\textit{M. V. Demina} et al., Qual. Theory Dyn. Syst. 21, No. 2, Paper No. 35, 35 p. (2022; Zbl 07489700)]. The methods used in this work may be used to study other physical ODE models in \(\mathbb{R}^2\).
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7614817
zbMATH Keywords: global analytic integrability
zbMATH Keywords: Liouville integrability
zbMATH Keywords: Puiseux integrability
reviewed by: Kaiyin Huang
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2022.103715
OpenAlex ID: W4292383677
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DOI: 10.1016/J.NONRWA.2022.103715
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description: scientific article; zbMATH DE number 7614817