23,360 research outputs found

    Boundary Value Problem for r2d2f/dr2+f=f3r^2 d^2 f/dr^2 + f = f^3 (III): Global Solution and Asymptotics

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    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

    Boundary Value Problem for r2d2f/dr2+f=f3r^2 {d^2 f/dr^2} + f = f^3 (I): Existence and Uniqueness

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    In this paper we study the equation r2d2f/dr2+f=f3r^2 {d^2 f/dr^2} + f = f^3 with the boundary conditions f(1)=0f(1)=0, f(∞)=1f(\infty)=1 and f(r)>0f(r) > 0 for 1<r<∞1<r<\infty. The existence of the solution is proved by using topological shooting argument. And the uniqueness is proved by variation method. Using the asymptotics of f(r)f(r) as rβ†’1r \to 1, in the following papers we will discuss the global solution for 0<r<∞0<r<\infty, and give explicit asymptotics of f(r)f(r) as rβ†’0r \to 0 and as rβ†’βˆžr \to \infty, and the connection formulas for the parameters in the asymptotics. Based on these results, we will solve the boundary value problem f(0)=0f(0) =0, f(∞)=1f(\infty) =1, which is the goal of this work. Once people discuss the regular solution of this equation, this boundary value problem must be considered. This problem is useful to study the Yang-Mills potential related equations, and the method used for this equation is applicible to other similar equations.Comment: 12 page
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