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
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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].
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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. |