Stability of the Travelling Front of a Decaying Brane

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.