A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis

Billingham, J.; Needham, D. J.

doi:10.1080/02681119108806105

Cited by 64 publications

(59 citation statements)

References 9 publications

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“…This contrasts with the fact that the critical wave speed c FKPP = 2 in the continuum limit (1) is determined locally, see [2,4,8,20,24,29]. In the corresponding travelling wave ODE, the origin is a stable node for c > 2, a degenerate stable node at c = 2 and a stable spiral when c < 2.…”

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 65%

“…Moreover, it is worth noting that Brunet and Derrida [9, section 4] also give physical arguments to show that the asymptotics in (4) are not restricted to their FKPP equation with cut-off in (2), but that the correction to the critical wave speed due to a cut-off in more general equations of a similar type will also be O((ln ε) −2 ). This type of behaviour should arise in numerical studies of the corresponding discrete models.…”

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 99%

“…The main results of this article are a rigorous, geometric derivation of the expansion in (4) for ϕ = (the Heaviside cut-off), as well as a proof of Brunet and Derrida's conjecture for a very general family of cut-off functions that do not even have to satisfy the condition that ϕ(u) 1 when u ε put forward in [9], cf (3). The proofs presented here are constructive and give precise information on the structure of the corresponding travelling waves.…”

Section: Introductionmentioning

confidence: 99%

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The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

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Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 65%

Section: Theorem 11 For Any Reaction-diffusion Equation Of the Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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“…It has been proved in [6] and [26] that there exists a minimal speed c 0 * (p) > 0 such that there exists a travelling wave V c (z) of (1.5)-(1.6) if and only if c ≥ c 0 * (p), and V c (z) satisfies V ′ c (z) < 0 for z ∈ R and * (p) were investigated in [22] and [10]. By applying similar phase plane analysis as in [6] and applying center manifold theorems, the above mentioned existence results for p degree Fisher equations are still valid for equation (1.5) with more general f (v) satisfying f ∈ C 2 ((0, 1]), f (0) = 0, f (v) > 0 for v ∈ (0, 1], and lim v→0 + f (v) v p = k 0 > 0 for some p > 1, (1.10) i.e. there exists a minimal speed c 0 * > 0 such that there exists a traveling wave V c (x − ct) satisfying (1.6) if and only if c ≥ c 0 * .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…As far as we know, there are no theoretical results on the stability of waves for the case when k is not small and d = 1 so far. For d = 1 and f (v) = v p with p > 1, the existence of traveling front solutions of the autocatalytic system (1.1) has been investigated in [1,6,9,10,15,19]. For d > 0 and f (v) = v 2 , it was shown in [6] that there exists a critical speed c * (d) such that (1.1) has traveling waves (U c (x − ct), V c (x − ct)) for any c ≥ c * (d), and the waves tend to (1, 0) algebraically as z → +∞ for c > c * (d).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

2013

Math. Model. Nat. Phenom.

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Abstract. This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems ut = duxx − uf (v), vt = vxx + uf (v) with d > 0 and d = 1, where f (v) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in C unif (R) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L1(R) norm and decays algebraically in C unif (R) norm, if the initial perturbation is in addition small in L1 norm.

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Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms

Malaguti

Marcelli

2002

Math. Nachr.

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We investigate wavefront solutions in a nonlinear system of two coupled reaction-diffusion equations with degenerate diffusivity:where t ≥ 0, x ∈ R, and D is a positive diffusion coefficient. This model, introduced by Kawasaki et al. (J. Theor. Biol. 188, 1997), describes the spatial-temporal dynamics of bacterial colonies b = b(x, t) and nutrients n = n(x, t) on agar plates. Kawasaki et al. provided numerical evidence for wavefronts, leaving the analytical confirmation of these solutions an open problem. We prove the existence of an infinite family of wavefronts parameterized by their wave speed, which varies on a closed positive halfline. We provide an upper bound for the threshold speed and a lower bound for it when D is sufficiently large. The proofs are based on several analytical tools, including the shooting method and the fixed-point theory in Fréchet spaces, to establish existence, and the central manifold theorem to ascertain uniqueness.

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A note on the properties of a family of travelling-wave solutions arising in cubic autocatalysis

Cited by 64 publications

References 9 publications

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

Travelling Wavefronts in Reaction-Diffusion Equations with Convection Effects and Non-Regular Terms

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