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

Abstract

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

11 pages, 2 fingures