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Monday, June 26, 2017

Wu-Yang monopole: numerical calculation

I have been reading the paper by Wu and Yang [1] in which they find the famous Wu-Yang monopole. In the paper there are solutions for three types of monopoles: one has an analytical form, which is the one most often quoted, but there are also two other monopoles with numerical solution only. In this post I use Python/numpy to perform numerical analysis on the latter solution. I use the same notation as in [1].
Wu and Yang obtain the following system of ordinary differential equations \begin{align} \frac{d\Phi}{d \xi} &= \psi\label{eq:20170625a}\\ \frac{d\psi}{d \xi} &= \psi + \Phi(\Phi^2-1)\label{eq:20170626a} \end{align} Here \xi is given by r = e^{\xi}, with r the distance to the origin. The right-hand side of \eqref{eq:20170625a}-\eqref{eq:20170626a} defines the vector field (d\Phi/d\xi, d\psi/d\xi) in the (\Phi, \psi) plane. Its integral curves can be seen in the next figure
The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The stationary points are marked in red.
I calculate the integral curve from the point (\Phi,\psi) = (0,0) to (1,0) using the numpy function solve_bvp [2].
The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The integral curve from the stationary point (0,0) to (1,0) is added in red.
\Phi(\xi) can be seen in the next graph. One sees that \Phi(\xi) \to 0 for \xi \to -\infty and \Phi(\xi) \to 1 for \xi \to +\infty
In the rest of this post I reproduce part of Table 1 in [1].