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. |

`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. |

Notice that if $(\Phi(\xi),\psi(\xi) )$ is a solution of \eqref{eq:20170625a}-\eqref{eq:20170626a}, then $(\Phi(\xi- \xi_0),\psi(\xi-\xi_0) )$ with $\xi_0$ a constant is also a solution. Translating to the $r$ coordinate, this means that if $\Phi(r)$ is a solution, the rescaled function $\Phi(r/c)$ is also a solution. Wu and Yang provide a table with numerical results on the function $\Phi(r)$ with asymptotic behaviour $\Phi(r) = 1 - 1/r +O(1/r^2)$ for $r \to \infty$. The solution plotted above has $\Phi(r) = 1 - c/r+O(1/r^2)$ for $r \to \infty$ with $c = 0.6233$ [3]. If I rescale my solution with $c$, I get the following table.

$\xi$ | $r/c$ | $\Phi(r/c)$ |
---|---|---|

5.066 | 9.880e+01 | 9.898e-01 |

2.866 | 1.095e+01 | 9.136e-01 |

1.666 | 3.297e+00 | 7.508e-01 |

0.566 | 1.098e+00 | 4.583e-01 |

-2.317 | 6.141e-02 | -9.229e-02 |

-5.954 | 1.617e-03 | 1.498e-02 |

-9.583 | 4.296e-05 | -2.442e-03 |

This agrees well with the results in Table 1 of Wu and Yang. I have not compared smaller values of $r$ because I have approximated the infinite interval $-\infty < \xi < + \infty$ by $-12\le \xi \le 8$.

References and comments

[1]
Wu and Yang, Some Solutions of the Classical Isotopic Gauge Field Equations, 1969

[2]
The Python code that I used can be found at this
link.

[3]
I have not estimated $c$ from the asymptotic behaviour of the solution that I found, but I have taken $c$ so that the difference between my solution and the one in Wu Yang is as small as possible.

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