Autocatalytic Reaction-diffusion System Research Articles

The minimal propagation speed of travelling waves for autocatalytic reaction-diffusion equations

This paper proves the existence of travelling waves for autocatalytic reaction-diffusion systems of a non-diffusive reactant and a diffusive autocatalyst where quadratic and cubic autocatalyses occur concurrently with the ratio 1 :k. We give the estimate of the minimal speed of travelling waves which is consistent with the result obtaind by S. Focant and Th. Gallay for the systems where a reactant and an autocatalyst are both diffusive. We further discuss the value of the parameterk which assures the validity of the heuristic argument employed by J. Murray and others.

Travelling waves in an ionic quadratic autocatalytic chemical system

The influence of a constant electric field on the travelling waves that are initiated in an ionic autocatalytic reaction-diffusion system with quadratic rate law is considered. The necessary conditions for the existence of minimum speed travelling waves are established and examples of such waves have been generated using numerical simulation. The behaviour of the system when the necessary conditions are not satisfied is also examined. This behaviour involves electrophoretic separation and reaction termination, each of which is discussed using examples obtained from numerical simulation.

Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system

We study a one-dimensional reaction-diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic ( A + B → 2 B) and a cubic ( A + 2 B → 3 B) autocatalysis. The parameters of this system are in the ratio D = D B D A of the diffusion constants of the reactant A and the autocatalyst B, and the relative activity k of the cubic reaction. First, for all values of D > 0 and k ≥ 0, we prove the existence of a family of propagating fronts (or travelling waves) describing the advance of the reaction. In particular, in the quadratic case k = 0, we recover the results of Billingham and Needham [Phil. Trans. R. Soc. London A 334 (1991) 1–24]. Then, if D is close to 1 and k is sufficiently small, we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted Sobolev spaces. This extends part of the results that are known for the scalar equation to which our system reduces when D = 1.

Travelling waves in an open quadratic autocatalytic chemical system

The travelling waves that are initiated in an autocatalytic reaction–diffusion system with quadratic rate law are considered. The system is modelled on the basis of a continuous-flow, unstirred reactor. The model is used to determine whether any of the complex structures reported for cubic autocatalytic reaction–diffusion systems can also be observed in the quadratic model. This is found not to be the case. The range of behaviour of the quadratic model is much less complex, with only front waves being initiated under the necessary conditions, which are established. There are, however, some unusual transient features to be found after the initial passage of the wave front.

Multiple steady states in a simple reaction–diffusion model with Michaelis–Menten (first-order Hinshelwood–Langmuir) saturation law: The limit of large separation in the two diffusion constants

The admissible multiple nonuniform steady states of a model bimolecular autocatalytic reaction–diffusion system with Michaelis–Menten (first-order Hinshelwood–Langmuir) saturation law are constructed in the case of large scale separation in the two diffusion constants. Both the Dirichlet and the Neumann problems are discussed in a one-dimensional geometry, and the corresponding bifurcation pictures are given.