Homoclinic bifurcations at the onset of pulse self-replication |
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Authors: | Arjen Doelman Tasso J Kaper |
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Institution: | a C.W.I. Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands b Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands c Department of Mathematics & Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA 02215, USA d Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands |
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Abstract: | We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u″=uv2, v″=v−uv2, which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches. |
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Keywords: | 35K45 35K57 35B25 35B32 35B35 35B40 34C30 34C37 92C15 92E20 |
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