The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff

Details The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff

2001-10-17

arxiv.org/abs/cond-mat/0110344v1

Phys. Rev. E 65, 057202 (2002)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 2 figures, submitted to Brief Reports, Phys. Rev. E

Scientific paper

10.1103/PhysRevE.65.057202

The concept of pulled fronts with a cutoff $\epsilon$ has been introduced to model the effects of discrete nature of the constituent particles on the asymptotic front speed in models with continuum variables (Pulled fronts are the fronts which propagate into an unstable state, and have an asymptotic front speed equal to the linear spreading speed $v^*$ of small linear perturbations around the unstable state). In this paper, we demonstrate that the introduction of a cutoff actually makes such pulled fronts weakly pushed. For the nonlinear diffusion equation with a cutoff, we show that the longest relaxation times $\tau_m$ that govern the convergence to the asymptotic front speed and profile, are given by $\tau_m^{-1} \simeq [(m+1)^2-1] \pi^2 / \ln^2 \epsilon$, for $m=1,2,...$.

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