Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (III): Global Solution and Asymptotics

Based on the results in the previous papers that the boundary value problem $y'' - y' + y = y^3, y(0) = 0, y(\infty) =1$ with the condition $y(x) > 0$ for $0<x<\infty$ has a unique solution $y^*(x)$, and $a^*= y^{*^{'}}(0)$ satisfies $0<a^*<1/4$, in this paper we show that $y'' - y' + y = y^3, -\infty < x < 0$, with the initial conditions $ y(0) = 0, y'(0) = a^*$ has a unique solution by using functional analysis method. So we get a globally well defined bounded function $y^*(x), -\infty < x < +\infty$. The asymptotics of $y^*(x)$ as $x \to - \infty$ and as $x \to +\infty$ are obtained, and the connection formulas for the parameters in the asymptotics and the numerical simulations are also given. Then by the properties of $y^*(x)$, the solution to the boundary value problem $r^2 f'' + f = f^3, f(0)= 0, f(\infty)=1$ is well described by the asymptotics and the connection formulas.