  We use open string field theory to study the dynamics of unstable branes in
the bosonic string theory, in the background of a generic linear dilaton. We
find a simple exact solution describing a dynamical interpolation between the
perturbative vacuum and the recently discovered nonperturbative tachyon vacuum.
In our solution, the open string tachyon increases exponentially along a null
direction, after which nonlinearities set in and cause the solution to
asymptote to a static state. In particular, the wild oscillations of the open
string fields which plague the time-like rolling tachyon solution are entirely
absent. Our model thus represents the first example proving that the true
tachyon vacuum of open string field theory can be realized as the endpoint of a
dynamical transition from the perturbative vacuum.
