Truncation effects in superdiffusive front propagation with Levy flights

A numerical and analytical study of the role of exponentially truncated Levy 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 not equal 0 this phenomenology prevails in the intermediate asymptotic regime (chi t)(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 phi similar to 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 ->infinity, 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)similar to v(*)-alpha/(lambda t), 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, phi similar to e(-nu x), and move at the constant velocity v=(gamma-lambda(alpha)chi)/nu.