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Chie Bing Wang

Publications and source records attributed to Chie Bing Wang.

5 recordsLinked to original sources
arXiv1999

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

In this paper we study the equation $r^2 {d^2 f/dr^2} + f = f^3$ with the boundary conditions $f(1)=0$, $f(\infty)=1$ and $f(r) > 0$ for $1<r<\infty$. The existence of the solution is proved by using topological shooting argument. And the uniqueness is proved by variation method. Using the asymptotics of $f(r)$ as $r \to 1$, in the following papers we will discuss the global solution for $0<r<\infty$, and give explicit asymptotics of $f(r)$ as $r \to 0$ and as $r \to \infty$, and the connection formulas for the parameters in the asymptotics. Based on these results, we will solve the boundary value problem $f(0) =0$, $f(\infty) =1$, 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.

math-ph↗
arXiv1999

Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (II): Connection Formula

In this paper, we study the analytic expansion in a small neighborhood of 0 in the complex plane for the solution to the equation $p dp/dz - p = z(z-1)(z-2)$ satisfying $p(z) = -z + O(z^2)$ as $z \to 0$. We show that the expansion is valid for $|z| \le s_0$, where $s_0 >1$. Then we get an explicit formula for $p(1)$ which is used to give the connection formula for the problem $r^2 f^{''} + f = f^3$, $f(1) = 0, f(\infty) = 1$.

math-ph↗
arXiv1999

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.

math-ph↗
arXiv1999

Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlevé II Equation

We consider the polynomials $ϕ_n(z)= κ_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrtλ (z+ 1/z)) dz/2 πi z$ on the unit circle in the complex plane. The leading coefficient $κ_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlevé II equation. The leading coefficient and second leading coefficient of $ϕ_n(z)$ can be expressed asymptotically in terms of the Painlevé II function.

solv-int↗