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Kao, Hsien-Ching; Knobloch, Edgar

Exact solutions of the cubic-quintic Swift-Hohenberg equation and their bifurcations. (English) Zbl 1294.35102

Dyn. Syst. 28, No. 2, 263-286 (2013).
Summary: Many systems of physical interest may be modelled by the bistable Swift-Hohenberg equation with cubic-quintic nonlinearity. We construct a two-parameter family of exact meromorphic solutions of the time-independent equation and use these to construct a one-parameter family of exact periodic solutions on the real line. These are of two types, differing in their symmetry properties, and are connected via an exact heteroclinic solution. We use these exact solutions as initial points for numerical continuation and show that some of these lie on secondary branches while others fall on isolas. The approach substantially enhances our understanding of the solution space of this equation.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q60 PDEs in connection with optics and electromagnetic theory
35B32 Bifurcations in context of PDEs
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs
34C23 Bifurcation theory for ordinary differential equations

Keywords:

exact meromorphic solutions; Laurent series; Swift-Hohenberg equation; bifurcations; numerical continuation

Software:

AUTO-07P; AUTO
PDFBibTeX XMLCite
\textit{H.-C. Kao} and \textit{E. Knobloch}, Dyn. Syst. 28, No. 2, 263--286 (2013; Zbl 1294.35102)
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