Pulse Dynamics in a ThreeComponent System: Existence Analysis
 Arjen Doelman,
 Peter van Heijster,
 Tasso J. Kaper
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Abstract
In this article, we analyze the threecomponent reactiondiffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activatorinhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype threecomponent system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary onepulse and twopulse solutions, and travelling onepulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for onepulse solutions, we analyze various bifurcations, including the saddlenode bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For twopulse solutions, we show that the third component is essential, since the reduced bistable twocomponent system does not support them. We also analyze the saddlenode bifurcation in which twopulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated sixdimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.
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 Title
 Pulse Dynamics in a ThreeComponent System: Existence Analysis
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Journal of Dynamics and Differential Equations
Volume 21, Issue 1 , pp 73115
 Cover Date
 20090301
 DOI
 10.1007/s1088400891252
 Print ISSN
 10407294
 Online ISSN
 15729222
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Threecomponent reactiondiffusion systems
 Onepulse solutions
 Travelling pulse solutions
 Twopulse solutions
 Geometric singular perturbation theory
 Melnikov function
 Primary: 35K55
 35B32
 34C37
 Secondary: 35K40
 Authors

 Arjen Doelman ^{(1)} ^{(2)}
 Peter van Heijster ^{(1)}
 Tasso J. Kaper ^{(3)}
 Author Affiliations

 1. Centrum voor Wiskunde en Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands
 2. Kortewegde Vries Instituut, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands
 3. Department of Mathematics & Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA, 02215, USA