Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2002-12-17
Physical Review Letters 91, 018302 (2003)
Nonlinear Sciences
Pattern Formation and Solitons
New version. Accepted for publication in Physical Review Letters
Scientific paper
10.1103/PhysRevLett.91.018302
The use of reaction-diffusion models rests on the key assumption that the underlying diffusive process is Gaussian. However, a growing number of studies have pointed out the prevalence of anomalous diffusion, and there is a need to understand the dynamics of reactive systems in the presence of this type of non-Gaussian diffusion. Here we present a study of front dynamics in reaction-diffusion systems where anomalous diffusion is due to the presence of asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator, whose fundamental solutions are Levy $\alpha$-stable distributions. Numerical simulation of the fractional Fisher-Kolmogorov equation, and analytical arguments show that anomalous diffusion leads to the exponential acceleration of fronts and a universal power law decay, $x^{-\alpha}$, of the tail, where $\alpha$, the index of the Levy distribution, is the order of the fractional derivative.
Carreras Benjamin A.
del-Castillo-Negrete Diego
Lynch V. E.
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