Figure 1

Asymptotic front velocity in the full stochastic model. (a) The solid line depicts the total mass $I(t)$ of type $A$ particles as a function of time. The system size is $L={10}^4$, the total number of $A$ and $B$ particles in the system is $200\times {10}^4$. Other parameters are $\gamma =1$ and $\mu =1.8$. A constant front velocity (constant $dI(t)/dt$) is attained asymptotically (dashed line). (b) Average shape of the wave front of type $A$ particles in the linear regime [$t=20$ in (a)] calculated from over 4000 realizations. The horizontal dashed line separates the exponential (dashed line) from the algebraic (dotted line) region. Note the log scale on the ordinate; below the dashed line the number of $A$ particles is of the order of unity. The inset shows the distance from the starting point $|X(t)|$ of a Lévy flight with $\mu =1.8$. The dashed line indicates the scaling $|X(t)|\sim t^{1/\mu }$.Reuse & Permissions

Figure 2

Propagation of wave fronts in superdiffusive systems with effective cutoff. (a) Location of the wave front $x^{*}(t)$ as a function of time $t$ for different particle numbers $N$ (solid lines) for $\mu =1.5$ and $\gamma =0.1$ in Eq. (7). Following initial transients the velocity $v=d{x}^{*}(t)/dt$ is constant (dashed lines) and increases with $N$. The inset depicts a magnification of the initial phase. The wave front accelerates exponentially but deviates from the mean-field dynamics ($\epsilon =0$, dashed line) after the transient period. (b) Shape of the wave front at exponentially increasing time steps $t={1.5}^m$ with $m=5,6,\dots ,20$ for $N={10}^4$. After a transient phase the shape remains unaltered, decays sharply for$1/N<u(x,t)<1$, and follows a power law $u(x,t)\sim x^{-(1+\mu )}$ for large $x$. The dashed line indicates the effective cutoff $\epsilon ={10}^{-4}$. The spatial extent in the numerical integration was $L=2^{23}\approx 8.38\times {10}^6$. Inset: Scaling behavior of the fronts’ profiles in the reference frame of the front. Arrows indicate the temporal direction. The curves approach a steady state characterized by an exponential decay in space, $\exp -\lambda x$. The position of the front $x^{*}(t)$ is given implicitly by $u\mathbf{(}{x}^{*}(t),t\mathbf{)}=0.05$.Reuse & Permissions

Figure 3

Asymptotic front velocity $v(N)$ of the effective cutoff model as a function of the particle number $N$ for different Lévy exponents $\mu =1.2$, 1.5, and 1.8 (circles, diamonds, and triangles, respectively). The dashed line indicates the scaling $v(N)\sim N^{1/\mu }$. The inset shows the calculated asymptotic front velocity of the full stochastic model for $\mu =1.8$.Reuse & Permissions