Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system

Merkin, J. H.; Needham, D. J.

doi:10.1007/bf00128907

Cited by 72 publications

(48 citation statements)

References 9 publications

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“…Properties of the travelling wave solutions. General properties of TWS for systems having kinetics of the form in (2.4)-(2.5), but with constant diffusion coefficients for both species, have been given in [2,10,13]. Some of these properties are preserved in the present case and we shall recall them without proof.…”

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confidence: 93%

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

Satnoianu¹,

Maini²,

Garduño³

et al. 2001

Discrete &Amp; Continuous Dynamical Systems - B

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Abstract. We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes, via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis.

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confidence: 93%

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

Satnoianu¹,

Maini²,

Garduño³

et al. 2001

Discrete &Amp; Continuous Dynamical Systems - B

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“…The same result was obtained by Liu et al [29] using the method of undetermined coefficients. The problem of selection of appropriate speed has been discussed in [30,31]. Fronts initiated on a compact support evolve to minimum velocity solutions [30].…”

Section: ∂ T U(x T) − D∂ 2 XX U(x T) = αU(x T)(1 − U(x T)/umentioning

confidence: 99%

“…The problem of selection of appropriate speed has been discussed in [30,31]. Fronts initiated on a compact support evolve to minimum velocity solutions [30]. The singular property, auto-Bcklund transformation, and analytical solutions, including some heteroclinic and homoclinic solutions of the Fisher equation, were obtained by Guo and Chen [32] via the expanded Painlevé analysis.…”

Section: ∂ T U(x T) − D∂ 2 XX U(x T) = αU(x T)(1 − U(x T)/umentioning

confidence: 99%

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

Kaushik¹

2007

Numerical Methods Partial

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This paper deals with the construction of a nonstandard numerical method to compute the travelling wave solutions of nonlinear reaction diffusion equations at high wave speeds. Related general properties are studied using the perturbation approximation. At high wave speed the perturbation parameter approaches to zero and the problem exhibits a multiscale character. That is, there are thin layers where the solution varies rapidly, while away from these layers the solution behaves regularly and varies slowly. Most of the conventional methods fail to capture this layer behavior. Thus, the quest for some new numerical techniques that may handle the travelling wave solutions at high wave speeds earns relevance. In this paper, one such parameter robust nonstandard numerical scheme is constructed, in the sense that its numerical solution converges in the maximum norm to the exact solution uniformly well for all finite wave speeds. To overcome the difficulty due to the nonlinearity, the problem is linearized using the quasilinearization process followed by nonstandard finite difference discretization. An extensive amount of analysis is carried out which uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiment is carried out in support of the predicted theory that validates computationally the theoretical results.

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“…This will lead to the concentration gradients, and then the advancing wave front of the reactant A (or the autocatalyst B) is thus generated and will propagate out from this initial zone. Mathematically, Billingham, Merkin, and Needham [25,5,6,28] have applied the numerical and asymptotical analysis to the system (1.1) with f (u) = k 1 u and K = 0 to confirm such a phenomenon of wave front propagation. We note that the rigorous proof of the existence of traveling wave front solutions to the system (1.1) with f (u) = k 1 u and K = 0 is given by Qi [30] and Chen and Qi [8] (see also Marion [24] and Ai and Huang [2] for the general reaction term f ).…”

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confidence: 99%

“…On the other hand, the analysis by Billingham, Merkin, and Needham [25,5,6,28] shows that no matter how small the amount of the autocatalyst B is introduced locally into the system (1.1) with f (u) = k 1 u and K = 0, traveling waves are always generated. This particular feature seems to be contrary to the fact that in many chemical systems, the initial input of an autocatalyst into the system must be above a threshold concentration for the initialization of traveling waves.…”

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confidence: 99%

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

Tsai

2010

Quart. Appl. Math.

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Abstract. The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the (m + 1)th order (A + mB → (m + 1)B) and a decay step of the same order (B → C) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor. Introduction.Since the pioneering works of Fisher [9] and of Kolmogorov, Petrovsky and Piskunov [23], traveling waves in a wide array of biological and chemical systems, based on the interaction between reaction and diffusion processes, have been extensively investigated. In many realistic systems, traveling waves can be initiated by a highly localized disturbance of a stable rest state (e.g., a highly localized increase in ion concentrations). A typical example is the Hodgkin-Huxley model [13] which describes the electrical activity across membranes of never cells. It is known that if a short, but sufficiently large, burst of simulating current is injected into a nerve axon, a so-called action potential wave can then be generated. In this paper, we will discuss a system which enjoys a similar phenomenon. Specifically, this system is governed by the following equations:

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Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system

Cited by 72 publications

References 9 publications

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay

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