with the conditions , and where the choice of a parabolic profile is arbitrary and can be changed. We also know that, if we can neglect the spatial part, the solution can be written down analytically as (see here and here):
being . Indeed, for we get the following pictures
The agreement is excellent confirming the fact that a strong coupling expansion is a gradient expansion. So, a large perturbation entering into a differential equation can be managed much in the same way one does for a small perturbation. In the case of ode look at this post.