Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach |
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Authors: | del-Castillo-Negrete D Carreras B A Lynch V E |
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Institution: | Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA. |
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Abstract: | The use of reaction-diffusion models rests on the key assumption that the diffusive process is Gaussian. However, a growing number of studies have pointed out the presence of anomalous diffusion, and there is a need to understand reactive systems in the presence of this type of non-Gaussian diffusion. Here we study front dynamics in reaction-diffusion systems where anomalous diffusion is due to asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator of order alpha, whose fundamental solutions are Levy alpha-stable distributions that exhibit power law decay, x(-(1+alpha)). Numerical simulations of the fractional Fisher-Kolmogorov equation and analytical arguments show that anomalous diffusion leads to the exponential acceleration of the front and a universal power law decay, x(-alpha), of the front's tail. |
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