# Convergence of perturbative solutions to a non-linear diff eq

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

\partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right)

We will look for solutions satisfying

\phi(\pm \infty) = \pm 1 \qquad \partial_{x}\phi\big|_{x=\pm \infty} = 0

Integrating once we find

\left(\partial_{x}\phi\right)^{2} = -\phi^{2} + \frac{\phi^{4}}{2} + 2\epsilon \left(\phi - \frac{\phi^{3}}{3}\right) + K

Using the boundary conditions we find that no solution exists for $\epsilon \neq 0$. For $\epsilon=0$ we find( assuming $\phi(0) = 0$)

\phi_{0} = \tanh\left( \frac{x}{\sqrt{2}} \right)

Now lets assume $\epsilon << 1$ and expand $\phi = \sum \epsilon^{j}\phi_{j}$ imposing the boundary conditions

\phi_{0}(\pm \infty) = \pm 1 \qquad \phi_{j}(\pm \infty) = 0

For j>0, and

\partial_{x}\phi_{i}\big|_{x=\pm \infty} = 0

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 $\phi_{5}$ without getting any wiser( I can write down the solutions if anybody asks but it takes a bit of time.)