Shift in the velocity of a front due to a cutoff

Abstract

We consider the effect of a small cut-off ε 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 cut-off ε is to select a single velocity which converges when ε → 0 to the one predicted by the marginal stability argument. For small ε, the shift in velocity has the form K(log ε) −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.

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

“…These numerical observations motivated Brunet and Derrida [9] to introduce the following modified or 'cut-off' FKPP equation:…”

“…One question addressed in [9] is precisely the effect of the cut-off function ϕ on the velocity c of the corresponding travelling fronts. In particular, for ϕ(u) = (u − ε), where denotes the Heaviside step function, it is shown in [9] that the front velocity c in (2) differs from the continuum wave speed c FKPP by a term that is logarithmic in ε, c ∼ 2 − π 2 (ln ε) 2 as ε → 0.…”

“…The motivation in [9] was that, for any fixed value of N , no particles are available to react at any point in the domain at which the concentration is less than ε = 1/N , and, hence, that the reaction function must be 'cut-off' there. One question addressed in [9] is precisely the effect of the cut-off function ϕ on the velocity c of the corresponding travelling fronts.…”

“…The motivation in [9] was that, for any fixed value of N , no particles are available to react at any point in the domain at which the concentration is less than ε = 1/N , and, hence, that the reaction function must be 'cut-off' there. One question addressed in [9] is precisely the effect of the cut-off function ϕ on the velocity c of the corresponding travelling fronts. In particular, for ϕ(u) = (u − ε), where denotes the Heaviside step function, it is shown in [9] that the front velocity c in (2) differs from the continuum wave speed c FKPP by a term that is logarithmic in ε, c ∼ 2 − π 2 (ln ε) 2 as ε → 0.…”

Propagation of CaMKII translocation waves in heterogeneous spiny dendrites

“…In other words, there is an effective lower cut-off within the leading edge of the front. In the case of nonlinear reaction diffusion equations, such discreteness effects have been shown to have a significant effect on the asymptotic velocity of a pulled front (Brunet and Derrida 1997;Panja 2004).…”