Reaction-diffusion models of decontamination

A contaminant, which also contains a polymer is in the form of droplets on a solid surface. It is to be removed by the action of a decontaminant, which is applied in aqueous solution. The contaminant is only sparingly soluble in water, so the reaction mechanism is that it slowly dissolves in the aqueous solution and then is oxidized by the decontaminant. The polymer is insoluble in water, and so builds up near the interface, where its presence can impede the transport of contaminant. In these circumstances, Dstl wish to have mathematical models that give an understanding of the process, and can be used to choose the parameters to give adequate removal of the contaminant. Mathematical models of this have been developed and analysed, and show results in broad agreement with the effects seen in experiments

Multi shocks in Reaction-diffusion models

It is shown, concerning equivalent classes, that on a one-dimensional lattice with nearest neighbor interaction, there are only four independent models possessing double-shocks. Evolution of the width of the double-shocks in different models is investigated. Double-shocks may vanish, and the final state is a state with no shock. There is a model for which at large times the average width of double-shocks will become smaller. Although there may exist stationary single-shocks in nearest neighbor reaction diffusion models, it is seen that in none of these models, there exist any stationary double-shocks. Models admitting multi-shocks are classified, and the large time behavior of multi-shock solutions is also investigated.Comment: 17 pages, LaTeX2e, minor revisio

An integration scheme for reaction-diffusion models

A detailed description and validation of a recently developed integration scheme is here reported for one- and two-dimensional reaction-diffusion models. As paradigmatic examples of this class of partial differential equations the complex Ginzburg-Landau and the Fitzhugh-Nagumo equations have been analyzed. The novel algorithm has precision and stability comparable to those of pseudo-spectral codes, but it is more convenient to employ for systems with quite large linear extention $L$. As for finite-difference methods, the implementation of the present scheme requires only information about the local enviroment and this allows to treat also system with very complicated boundary conditions.Comment: 14 page, Latex - 4 EPS Figs - Submitted to Int. J. Mod. Phys.

Reaction-diffusion models for biological pattern formation

We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction-terms for several illustrative examples. After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. Comparisons are drawn between solution behaviour on growing domains and recent results on self-replicating patterns on domains of fixed size

Reaction Diffusion Models in One Dimension with Disorder

We study a large class of 1D reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g. of several species) undergo diffusion with random local bias (Sinai model) and react upon meeting. We obtain the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of exponents which characterize the convergence towards the asymptotic states. For reactions with several asymptotic states, we analyze the dynamical phase diagram and obtain the critical exponents at the transitions. We also study persistence properties for single particles and for patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories ($\theta$) or the thermally averaged packets ($\bar{\theta}$). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process $A+A \to \emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents $\delta(r)$ and $\psi(r)$ characterizing the survival up to time t of a domain without any merging or with mergings respectively, and $\delta_A(r)$ and $\psi_A(r)$ characterizing the survival up to time t of a particle A without any coalescence or with coalescences respectively. $\bar{\theta}, \psi, \delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). Additional disorder in the reaction rates, as well as some open questions, are also discussed.Comment: 54 pages, Late

The phase transition of triplet reaction-diffusion models

The phase transitions classes of reaction-diffusion systems with multi-particle reactions is an open challenging problem. Large scale simulations are applied for the 3A -> 4A, 3A -> 2A and the 3A -> 4A, 3A->0 triplet reaction models with site occupation restriction in one dimension. Static and dynamic mean-field scaling is observed with signs of logarithmic corrections suggesting d_c=1 upper critical dimension for this family of models.Comment: 4 pages, 4 figures, updated version prior publication in PR