  The dynamics (in light-cone time) of the tachyon on an unstable brane in the
background of a dilaton linear along a null coordinate is a non-local
reaction-diffusion type equation, which admits a travelling front solution. We
analyze the (in-)stability of this solution using linearized perturbation
theory. We find that the front solution obtained in singular perturbation
method is stable. However, these inhomogenous solutions (unlike the homogenous
solution) also have Lyapunov exponents corresponding to unstable modes around
the (meta-)stable vacuum.
