Precursors in Front Propagation

We investigate the dynamical construction of the leading edge of propagating fronts. Whereas the steady-state front is typically an exponential, far ahead of the front, the front falls off much faster, in a fashion determined by the Green's function of tile problem. We show that there is a universal transition Tom the steady-state exponential front to a Gaussian falloff. The transition region is of width t1/2 , and moves out ahead of the front at a constant velocity greater than the steady-state front speed. This Gaussian front then is in general modified even further ahead of the front to match onto the expected Green's function behavior. We demonstrate this in the case of the Ginzburg-Landau and Korteweg-De Vries equations. We also discuss the relevance of this mechanism for velocity selection in the Fisher equation