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Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations. (English) Zbl 1029.34036
Summary: The authors establish the existence of homoclinic solutions to 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 Y. Chen and P. J. McKenna [ibid. 136, No. 2, 325-355 (1997; Zbl 0879.35113)], the authors give a positive answer to an open question raised by them.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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