Lévy飞行超扩散波前传播中的截断效应
Truncation effects in superdiffusive front propagation with Lévy flights
摘要：
对反应扩散系统中指数截断Lévy飞行在前沿超扩散传播中的作用进行了数值和分析研究。该研究是基于一个变化的费舍尔-Kolmogorov方程的扩散算子被替换为$λ$ -截断分数阶导数$α$其中$1/λ$是特征截断长度尺度。对于$λ=0$，没有截断，前沿呈现指数加速和代数衰减尾部。结果表明，$λ\neq 0$这种现象普遍存在于中间渐近制度$(χt)^{1/α} \ll x \ll 1/λ$ $χ$是扩散常数。在中间渐近状态之外，即对于$x > 1/λ$，锋的尾部表现出缓和衰减$ϕ\sim e^{-λx}/x^{(1+α)} $，加速度是瞬时的，并且锋速度$v_L$以$t\to \infty$接近终速$v_* = (γ- λ^αχ)/λ$，其中假定$γ> λ^αχ$和$γ$表示反应动力学的增长速率。然而，这个过程的收敛是代数的，$v_L \sim v_* - α/(λt)$，这是非常缓慢的扩散（高斯）的情况下观察到的指数收敛相比。一个过度截断制度，其中的特征截断长度尺度是短于初始条件，$1/ν$的衰减的长度尺度，也被确定。在这种极端的状态下，锋面呈现指数尾$ϕ\sim e^{-νx}$，并以恒定速度$v=(γ- λ^αχ)/ν$移动。
英文摘要：
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, $v=(γ- λ^αχ)/ν$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
