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An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations

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Abstract

In recent years, methods have been developed to study the existence, stability and bifurcations of pulses in singularly perturbed reaction–diffusion equations in one space dimension, in the context of a number of explicit model problems, such as the Gray–Scott and the Gierer–Meinhardt equations. Although these methods are in principle of a general nature, their applicability a priori relies on the characteristics of these models. For instance, the slow reduced spatial problem is linear in the models considered in the literature. Moreover, the nonlinearities in the fast reduced spatial problem are of a very specific, polynomial, nature. These properties are crucially used, especially in the stability and bifurcation analysis. In this paper, we present an explicit theory for pulses in two-component singularly perturbed reaction–diffusion equations that significantly extends and generalizes existing methods.

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Acknowledgments

This research was funded by NWO through the VIDI-grant ‘Formation of singularities in natural systems’, awarded to V. Rottschäfer, Universiteit Leiden, The Netherlands.

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  1. Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA , Leiden, The Netherlands

    Arjen Doelman & Frits Veerman

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  1. Arjen Doelman
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  2. Frits Veerman
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Correspondence to Arjen Doelman.

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Dedicated to Klaus Kirchgässner, in gratitude for his inspiration and stimulation.

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Doelman, A., Veerman, F. An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations. J Dyn Diff Equat 27, 555–595 (2015). https://doi.org/10.1007/s10884-013-9325-2

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  • DOI: https://doi.org/10.1007/s10884-013-9325-2

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