  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.
