Homoclinic Solutions for Swift-Hohenberg and Suspension Bridge Type Equations |
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Authors: | Didier SmetsJan Bouwe van den Berg |
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Affiliation: | a Université Catholique de Louvain, Département de Mathématiques, 2 Chemin du Cyclotron, 1348, Louvain-la-Neuve, Belgiumf1smets@ann.jussieu.frf1b Division of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdomf2jan.bouwe@nottingham.ac.ukf2 |
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Abstract: | We establish the existence of homoclinic solutions for a class of fourth-order equations which includes the Swift-Hohenberg model and the suspension bridge equation. In the first case, the nonlinearity has three zeros, corresponding to a double-well potential, while in the second case the nonlinearity is asymptotically constant on one side. The Swift-Hohenberg model is a higher-order extension of the classical Fisher-Kolmogorov model. Its more complicated dynamics give rise to further possibilities of pattern formation. The suspension bridge equation was studied by Chen and McKenna (J. Differential Equations136 (1997), 325-355); we give a positive answer to an open question raised by the authors. |
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