Superfast front propagation in reactive systems with non-Gaussian diffusion

Abstract

We study a reaction diffusion system where we consider a non-gaussian process instead of a standard diffusion. If the process increments follow a probability distribution with tails approaching to zero faster than a power law, the usual qualitative behaviours of the standard reaction diffusion system, i.e., exponential tails for the reacting field and a constant front speed, are recovered. On the contrary if the process has power law tails, also the reacting field shows power law tail and the front speed increases exponentially with time. The comparison with other reaction-transport systems which exhibit anomalous diffusion shows that, not only the presence of anomalous diffusion, but also the detailed mechanism, is relevant for the front propagation.Pacs: 47.70.Fw

Fronts in anomalous diffusion–reaction systems

“…Therefore, travelling wave solutions of the form u(x − ct) are not expected. [49]. del Castillo-Negrete et al [39] investigated the phenomenon of the front acceleration in the framework of a fractional superdiffusion-reaction equation with a fully asymmetric superdiffusion operator, (2.12) which corresponds to equation (2.6) with θ = −1 and a rescaled spatial variable, for a front between the stable phase, u(−∞) = 1, and an unstable phase, u(∞) = 0. del Castillo-Negrete et al considered the evolution of a small disturbance governed by a linearized equation…”

Front Dynamics in Reaction-Diffusion Systems with Levy Flights: A Fractional Diffusion Approach

“…More recently, superfast front propagation in reactive systems with non-Gaussian diffusion was discussed in Ref. [14]. The model studied in Ref.…”

“…The model studied in Ref. [14] consisted of a time-discrete reaction system coupled to a superdiffusive Levy process described by an integral operator with an algebraic decaying propagator.…”

Asymptotic Scaling Symmetries for Nonlinear Pdes

“…This includes processes with non-independent but finite variance increments (coloured noise), and processes with independent increments but infinite variance (Levy flights); see e.g. [30,31] again for examples and detailed numerical studies. The relevant point is that, in order to apply our approach, the considered stochastic processes should present an asymptotic probability distribution (starting with sufficiently localized initial distribution) of the form…”