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. |