Homoclinic Solutions for Swift–Hohenberg and Suspension Bridge Type Equations

✍ Scribed by Didier Smets; Jan Bouwe van den Berg

• Book ID: 102589179
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 194 KB
• Volume: 184
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.2001.4135

✦ Synopsis

We establish the existence of homoclinic solutions for a class of fourthorder 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 Equations 136 (1997), 325-355); we give a positive answer to an open question raised by the authors.