  We consider solitonic solutions of the DBI tachyon effective action for a
non-BPS brane. When wrapped on a circle, these solutions are regular and have a
finite energy. We show that in the decompactified limit, these solitons give
Sen's infinitely thin finite energy kink -- interpreted as a BPS brane --
provided that some conditions on the potential hold. In particular, if for
large $T$ the potential is exponential, $V = e^{-T^a}$, then Sen's solution is
only found for $a<1$. For power-law potentials $V = 1/T^b$, one must have
$b>1$. If these conditions are not satisfied, we show that the lowest energy
configuration is the unstable tachyon vacuum with no kinks. We examine the
stability of the solitons and the spectrum of small perturbations.
