The Existence of Solitary Traveling Wave Solutions of a Model of the Belousov-Zhabotinskii Reaction

Field, Richard J.; Troy, William C.

doi:10.1137/0137042

Cited by 21 publications

(4 citation statements)

References 27 publications

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“…Since this celebrated pioneering paper, there has been a great amount of work on the questions of existence, uniqueness, or stability properties of planar traveling fronts for different types of reaction terms f (u) arising in combustion or biological models; see, for example, Aronson and Weinberger [1], Fife and McLeod [34], Johnson and Nachbar [55], and Kanel [59]. Many papers have also been devoted to the study of planar traveling fronts for systems of one-dimensional diffusion-reaction equations [8,15,20,29,32,38,60,77,81,87,98]. The results have shown either some differences from the case of single equations or some analogies.…”

Section: One-dimensional Resultsmentioning

confidence: 99%

Front propagation in periodic excitable media

Berestycki

Hamel

2002

Comm Pure Appl Math

346

546

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This paper is devoted to the study of pulsating traveling fronts for reactiondiffusion-advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating traveling fronts generalizes that of traveling fronts for planar or shear flows.Various existence, uniqueness, and monotonicity results are proved for two classes of reaction terms. First, for a combustion-type nonlinearity, it is proved that the pulsating traveling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity that arises either in combustion or biological models. Then, the set of possible speeds is a semiinfinite interval, closed and bounded from below. For each possible speed, there exists a pulsating traveling front that is increasing in time. This result extends the classical Kolmogorov-Petrovsky-Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes.

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Section: One-dimensional Resultsmentioning

confidence: 99%

Front propagation in periodic excitable media

Berestycki

Hamel

2002

Comm Pure Appl Math

346

546

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“…Travelling waves in clock reactions were first described by Luther (1906) and models based on similar kinetics to those employed here have been widely used since Fisher (1937) and Kolmogorov et al (1937). A quantitative application of cubic autocatalytic rate laws in these circumstances to the iodate-arsenite reaction has been given by Hanna et al (1982) and Saul & Showalter (198 model has been used by Field & Troy (1979) Reusser & Field (1979 for the Belousov-Zhabotinskii reaction (Field & Noyes 1974), Showalter et al (1979) and Kopell & Howard (1981). In biological systems, travelling-wave solutions in nerve conduction have been modelled via the Hodgkin-Huxley (1952) equations or their simplified forms such as the Fitzhugh-Nagumo equation (Fitzhugh 1969;Nagumo et al 1962).…”

Section: Discussionmentioning

confidence: 99%

Multiple stationary states, sustained oscillations and transient behaviour in autocatalytic reaction-diffusion equations

Kay

Scott

1988

Proc. R. Soc. Lond. A

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The behaviour of the prototype cubic autocatalator A + 2B → 3B rate = k 1 ab 2 , B → C rate = k 2 b , when reaction is coupled with diffusion from a surrounding reservoir of constant composition is investigated. For indefinitely stable catalysts (i. e. for k 2 = 0) the model exhibits ignition, extinction (wash-out) and hysteresis. The range of conditions over which multiple stationary states are found decreases as the concentration of the autocatalyst b ex in the reservoir increases. Finally ignition and extinction points merge in a cusp catastrophe with the consequent loss of multiplicity. Away from the critical points, the ultimate approach to the stationary state is, in general, governed by an exponential decay. The characteristic relaxation time for this approach lengthens as ignition or extinction points are approached, thus displaying ‘slowing down’. Eventually, non-exponential time-dependences are also found. With finite catalyst lifetimes ( k 2 > 0), the dependence of the stationary-state composition on the diffusion rate or the size of the reaction zone shows more complex patterns. Five qualitatively different responses can be found: (i) unique, (ii) isola, (iii) breaking wave, (iv) mushroom and (v) breaking wave + isola. The stationary-state profile for the distribution of the autocatalytic species B now allows multiple internal extrema (the onset of dissipative structures). The cubic autocatalator also provides the simplest, yet chemically consistent, example of temporal and spatial oscillations in a reaction-diffusion system.

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“…Using (8) and (9) we want to show there is a C1 function <i> (8) Then for e sufficiently small, 3//8<f> *■ 0, provided X3p(X3) sin0cos<i> -X2p(X2) sin<|> # 0.…”

Section: W=r*\(pu Qu Ru S)mentioning

confidence: 99%

“…Previous shooting arguments to establish the existence of traveling wave solutions in reaction-diffusion equations, e.g. in [8,15,23], have been "disconnection" arguments. That is, the unstable manifold at the target point is 1-dimensional, while the stable manifold is (n -l)-dimensional.…”

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confidence: 99%

Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in 𝑅⁴

Dunbar¹

1984

Trans. Amer. Math. Soc.

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Abstract. We establish the existence of traveling wave solutions for a reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction. The waves are of transition front type, analogous to the solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction-diffusion equation. The waves discussed here are not necessarily monotone. There is a speed c* > 0 such that for c > c* there is a traveling wave moving with speed c. The proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a Liapunov function to guarantee the existence of the traveling wave solution. The traveling wave solution is equivalent to a heteroclinic orbit in 4-dimensional phase space.

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The Existence of Solitary Traveling Wave Solutions of a Model of the Belousov-Zhabotinskii Reaction

Cited by 21 publications

References 27 publications

Front propagation in periodic excitable media

Front propagation in periodic excitable media

Multiple stationary states, sustained oscillations and transient behaviour in autocatalytic reaction-diffusion equations

Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in 𝑅⁴

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