Truncation effects in superdiffusive front propagation with Lévy flights

Abstract

A numerical and analytical study of the role of exponentially truncated Lévy flights in the superdiffusive propagation of fronts in reaction-diffusion systems is presented. The study is based on a variation of the Fisher-Kolmogorov equation where the diffusion operator is replaced by a $λ$-truncated fractional derivative of order $α$ where $1/λ$ is the characteristic truncation length scale. For $λ=0$ there is no truncation and fronts exhibit exponential acceleration and algebraic decaying tails. It is shown that for $λ\neq 0$ this phenomenology prevails in the intermediate asymptotic regime $(χt)^{1/α} \ll x \ll 1/λ$ where $χ$ is the diffusion constant. Outside the intermediate asymptotic regime, i.e. for $x > 1/λ$, the tail of the front exhibits the tempered decay $ϕ\sim e^{-λx}/x^{(1+α)} $, the acceleration is transient, and the front velocity, $v_L$, approaches the terminal speed $v_* = (γ- λ^αχ)/λ$ as $t\to \infty$, where it is assumed that $γ> λ^αχ$ with $γ$ denoting the growth rate of the reaction kinetics. However, the convergence of this process is algebraic, $v_L \sim v_* - α/(λt)$, which is very slow compared to the exponential convergence observed in the diffusive (Gaussian) case. An over-truncated regime in which the characteristic truncation length scale is shorter than the length scale of the decay of the initial condition, $1/ν$, is also identified. In this extreme regime, fronts exhibit exponential tails, $ϕ\sim e^{-νx}$, and move at the constant velocity, $v=(γ- λ^αχ)/ν$.