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 $\lambda$-truncated fractional derivative of order $\alpha$ where $1/\lambda$ is the characteristic truncation length scale. For $\lambda =0$ there is no truncation and fronts exhibit exponential acceleration and algebraic decaying tails. It is shown that for $\lambda \ne 0$ this phenomenology prevails in the intermediate asymptotic regime $(\chi t{)}^{1/\alpha }\ll x\ll 1/\lambda$ where $\chi$ is the diffusion constant. Outside the intermediate asymptotic regime, i.e. for $x>1/\lambda$, the tail of the front exhibits the tempered decay $\varphi \sim e^{-\lambda x}/x^{(1+\alpha )}$, the acceleration is transient, and the front velocity, $v_L$, approaches the terminal speed $v_{\ast }=(\gamma -{\lambda }^{\alpha }\chi )/\lambda$ as $t\to \infty$, where it is assumed that $\gamma >{\lambda }^{\alpha }\chi$ with $\gamma$ denoting the growth rate of the reaction kinetics. However, the convergence of this process is algebraic, $v_L\sim v_{\ast }-\alpha /(\lambda t)$, 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, $1/\nu$, is also identified. In this extreme regime, fronts exhibit exponential tails, $\varphi \sim e^{-\nu x}$, and move at the constant velocity, $v=(\gamma -{\lambda }^{\alpha }\chi )/\nu$.

Accepted for publication in Phys. Rev. E (Feb. 2009)

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