Boundary Value Problem for r2d2f/dr2+f=f3r^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 yβ€²β€²βˆ’yβ€²+y=y3,y(0)=0,y(∞)=1y'' - y' + y = y^3, y(0) = 0, y(\infty) =1 with the condition y(x)>0y(x) > 0 for 0<x<∞0<x<\infty has a unique solution yβˆ—(x)y^*(x), and aβˆ—=yβˆ—β€²(0)a^*= y^{*^{'}}(0) satisfies 0<aβˆ—<1/40<a^*<1/4, in this paper we show that yβ€²β€²βˆ’yβ€²+y=y3,βˆ’βˆž<x<0y'' - y' + y = y^3, -\infty < x < 0, with the initial conditions y(0)=0,yβ€²(0)=aβˆ— y(0) = 0, y'(0) = a^* has a unique solution by using functional analysis method. So we get a globally well defined bounded function yβˆ—(x),βˆ’βˆž<x<+∞y^*(x), -\infty < x < +\infty. The asymptotics of yβˆ—(x)y^*(x) as xβ†’βˆ’βˆžx \to - \infty and as xβ†’+∞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 yβˆ—(x)y^*(x), the solution to the boundary value problem r2fβ€²β€²+f=f3,f(0)=0,f(∞)=1r^2 f'' + f = f^3, f(0)= 0, f(\infty)=1 is well described by the asymptotics and the connection formulas.Comment: 11 pages, 2 fingure

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