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 algebraically decaying tails. It is shown that for $\lambda \ne 0$ this phenomenology prevails in the intermediate asymptotic regime ${\left(\chi t\right)}^{1∕\alpha }⪡x⪡1∕\lambda$ 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 $v_{*}=(\gamma -{\lambda }^{\alpha }\chi )∕\lambda$ 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, $v_L\sim v_{*}-\alpha ∕\left(\lambda t\right)$, which is very slow compared to the exponential convergence observed in the diffusive (Gaussian) case. An overtruncated 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 $v=(\gamma -{\lambda }^{\alpha }\chi )∕\nu$.

• Received 18 December 2008

DOI:https://doi.org/10.1103/PhysRevE.79.031120