Emden-Fowler方程的非线性稳定性分析
Nonlinear stability analysis of the Emden-Fowler equation
摘要：
本文定性地研究了半线性椭圆型方程$Δu + u^n = 0$在正实直线上的径向解，其中$u(0)=1$和$u'(0)=0$在正实直线上的径向解称为Emden-Fowler或Lane-Emden方程.这个方程在牛顿天体物理学中非常重要，常数$n$称为多方指数。通过引入一组新的变量，Emden-Fowler方程可以写成两个常微分方程的自治系统，可以使用线性和非线性稳定性分析来分析。我们使用线性稳定性分析，Jacobi稳定性分析（Kosambi-Cartan-Chern理论）和李雅普诺夫函数方法进行稳定性研究。根据$n$的值，这些不同的方法产生不同的结果。我们确定了$n$的参数范围，其中所有三种方法都意味着稳定性。
英文摘要：
In this paper we qualitatively study radial solutions of the semilinear elliptic equation $Δu + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant $n$ is called the polytropic index. By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of $n$ these different methods yield different results. We identify a parameter range for $n$ where all three methods imply stability.
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