Shift in the velocity of a front due to a cutoff

Abstract

We consider the effect of a small cutoff $\ensuremath{\varepsilon}$ on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff $\ensuremath{\varepsilon}$ is to select a single velocity that converges when $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$ to the one predicted by the marginal stability argument. For small $\ensuremath{\varepsilon}$, the shift in velocity has the form $K(\mathrm{ln}\ensuremath{\varepsilon}{)}^{\ensuremath{-}2}$ and our prediction for the constant $K$ agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite-size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.