Strum
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Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it.
We have the equation
\begin{equation}
\partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right)
\end{equation}
We will look for solutions satisfying
\begin{equation}
\phi(\pm \infty) = \pm 1 \qquad \partial_{x}\phi\big|_{x=\pm \infty} = 0
\end{equation}
Integrating once we find
\begin{equation}
\left(\partial_{x}\phi\right)^{2} = -\phi^{2} + \frac{\phi^{4}}{2} + 2\epsilon \left(\phi - \frac{\phi^{3}}{3}\right) + K
\end{equation}
Using the boundary conditions we find that no solution exists for [itex]\epsilon \neq 0[/itex]. For [itex]\epsilon=0[/itex] we find( assuming [itex]\phi(0) = 0[/itex])
\begin{equation}
\phi_{0} = \tanh\left( \frac{x}{\sqrt{2}} \right)
\end{equation}
Now let's assume [itex]\epsilon << 1[/itex] and expand [itex]\phi = \sum \epsilon^{j}\phi_{j}[/itex] imposing the boundary conditions
\begin{equation}
\phi_{0}(\pm \infty) = \pm 1 \qquad \phi_{j}(\pm \infty) = 0
\end{equation}
For j>0, and
\begin{equation}
\partial_{x}\phi_{i}\big|_{x=\pm \infty} = 0
\end{equation}
For all i.
Now what will the higher and higher order terms signify? And will this series ever converge towards anything( either stationary or periodic)? I worked out the expansion up to [itex]\phi_{5}[/itex] without getting any wiser( I can write down the solutions if anybody asks but it takes a bit of time.)
We have the equation
\begin{equation}
\partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right)
\end{equation}
We will look for solutions satisfying
\begin{equation}
\phi(\pm \infty) = \pm 1 \qquad \partial_{x}\phi\big|_{x=\pm \infty} = 0
\end{equation}
Integrating once we find
\begin{equation}
\left(\partial_{x}\phi\right)^{2} = -\phi^{2} + \frac{\phi^{4}}{2} + 2\epsilon \left(\phi - \frac{\phi^{3}}{3}\right) + K
\end{equation}
Using the boundary conditions we find that no solution exists for [itex]\epsilon \neq 0[/itex]. For [itex]\epsilon=0[/itex] we find( assuming [itex]\phi(0) = 0[/itex])
\begin{equation}
\phi_{0} = \tanh\left( \frac{x}{\sqrt{2}} \right)
\end{equation}
Now let's assume [itex]\epsilon << 1[/itex] and expand [itex]\phi = \sum \epsilon^{j}\phi_{j}[/itex] imposing the boundary conditions
\begin{equation}
\phi_{0}(\pm \infty) = \pm 1 \qquad \phi_{j}(\pm \infty) = 0
\end{equation}
For j>0, and
\begin{equation}
\partial_{x}\phi_{i}\big|_{x=\pm \infty} = 0
\end{equation}
For all i.
Now what will the higher and higher order terms signify? And will this series ever converge towards anything( either stationary or periodic)? I worked out the expansion up to [itex]\phi_{5}[/itex] without getting any wiser( I can write down the solutions if anybody asks but it takes a bit of time.)