Speed of Fronts of the Reaction-Diffusion Equation

Benguria, Rafael D.; Depassier, M. C.

doi:10.1103/physrevlett.77.1171

Cited by 93 publications

(110 citation statements)

References 20 publications

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“…For intermediate ν we must resort to numerical calculations. The approach is a generalization of a method developed for other nonlinear problems [19,20,21]. This first approach enabled us to show that in cases of cubic F with a unique limit cycle, its location can be obtained from a minimum principle [22].…”

Section: Introductionmentioning

confidence: 99%

Variational approach to a class of nonlinear oscillators with several limit cycles

Depassier

Mura

2001

Phys. Rev. E

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We study limit cycles of nonlinear oscillators described by the equationẍ + νF (ẋ) + x = 0 with F an odd function. Depending on the nonlinearity, this equation may exhibit different number of limit cycles. We show that limit cycles correspond to relative extrema of a certain functional.Analytical results in the limits ν → 0 and ν → ∞ are in agreement with previously known criteria.For intermediate ν numerical determination of the limit cycles can be obtained.

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Section: Introductionmentioning

confidence: 99%

Variational approach to a class of nonlinear oscillators with several limit cycles

Depassier

Mura

2001

Phys. Rev. E

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“…Theorem 37 expands on the work of Benguria and Depassier [33,[35][36][37][38] and of Benguria, Cisternas and Depassier [32]. Indeed, the most essential ideas behind the proof of part (a) can be found in [32,33,35,36,38] and part (b) in [37].…”

Section: Elementary Analysis Shows That This Occurs If and Only Ifmentioning

confidence: 98%

“…Indeed, the most essential ideas behind the proof of part (a) can be found in [32,33,35,36,38] and part (b) in [37].…”

Section: Elementary Analysis Shows That This Occurs If and Only Ifmentioning

confidence: 99%

“…In other words, it is possible to view the integral equation (1.9) as a description of travelling-wave solutions of (1.1) in their phase-space. The relationship between travelling-wave solutions of equations of the type (1.1) and the ordinary differential equation (1.11) has been noted and utilized by previous authors [20][21][22][23][32][33][34][35][36][37][38]61,90,93,96,97,159,162,172,191,192,213,222,246,247,250,254,266,268,270,282], and for a related problem in [153,154,282].…”

Section: C(r)a (R) G(r) Drmentioning

confidence: 99%

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Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

170

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The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation.

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“…In general, the minimum wave speed c * of travelling wave solutions of (2.1) is unknown (although bounds have been derived: see [3,26]. However for the special case…”

Section: The Form Of Smooth-front Wavesmentioning

confidence: 99%

On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

Sherratt

2010

Math. Model. Nat. Phenom.

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Abstract. Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front waves with speeds close to that of the (unique) sharp-front solution, for the particular equationUsing singular perturbation theory, the author derives an asymptotic approximation to the wave, which gives valuable information about the structure of smooth-front solutions. The approximation compares well with numerical results.Key words: nonlinear diffusion, travelling waves, sharp front, smooth front AMS subject classification: 35K57, 35K65, 92B05 Nonlinear Diffusion in Biological ModelsThe first applications of reaction-diffusion equations in biology [8,48] used standard linear diffusion to model population spread. However, it is now clear that in a number of biological contexts, motility varies with population density, requiring nonlinear diffusion terms. This was first realised in ecological applications [12,13,45], and density-dependent dispersal is now a common feature *

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Speed of Fronts of the Reaction-Diffusion Equation

Abstract: We study the speed of propagation of fronts for the scalar reaction-diffusion

Cited by 93 publications

References 20 publications

Variational approach to a class of nonlinear oscillators with several limit cycles

Variational approach to a class of nonlinear oscillators with several limit cycles

Travelling Waves in Nonlinear Diffusion-Convection Reaction

On the Form of Smooth-Front Travelling Waves in a Reaction-Diffusion Equation with Degenerate Nonlinear Diffusion

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