Dynamics and stability of light-like tachyon condensation

Recently, Hellerman and Schnabl considered the dynamics of unstable d-branes in the background of a linear dilation. Remarkably, they were able to construct light-like tachyon solutions which interpolate smoothly between the perturbative and nonpertubative vacua, without undergoing the wild ocillations that plague time-like solutions. In their analysis, however, the full structure of the initial value problem for the nonlocal dynamical equations was no considered. In this paper, therefore, we reexamine the nonlinear dynamics of light-like tachyon condensation using a combination of numerical and analytical techniques. We find that for a p-adic string the monotonic behaviour obtained previously relied on a special choice of initial conditions near the unstable maximum. for generic initial conditions the wild ocillations come back to haunt us. Interestingly, we find an "island of stability" in initial condition space that leads to sensible evolution at late times. for the string field theory case, on the other hand, we find that the evolution is completely stable for generic choices of initial data. This provides an explicit example of a string theoretic system that admits infinitely many initial data but is nevertheless nonperturbatively stable. Qualitatively similar dynamics are obtained in nonlocal cosmologies where the Hubble damping plays a role very analogous to the dilation gradient.