I guess I'll answer this question since I found it out.
It turned out to work better using the EOM for α. The main issue here is as t blows up, the derivative (after converting to first order system of equations) begins to blow up making various RK-methods unstable.
We first can try fixing the b.c. at something like for l big but it turns out, here, to model the profile (amplitude) I want to. It turns out setting the upper limit to somehting just under 1 works well, then stretch the solution for by a tanh.