Localized Structures in the Multi-dimensional Swift-Hohenberg Equation

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Title
Localized Structures in the Multi-dimensional Swift-Hohenberg Equation
Contributors
McCalla, Scott G (creator)
Sandstede, Bjorn (Director)
Mallet-Paret, John (Reader)
Scheel, Arnd (Reader)
Brown University. Applied Mathematics (sponsor)
Copyright Date
2011
Abstract
This goal of this thesis is to understand patterns in the Swift{Hohenberg equation. Thepatterns studied are localized, stationary and radially symmetric in dimensions one throughthree. The emphasis is placed on the existence of these structures through numericalevidence and analytic proofs.The bifurcation structure of localized stationary radial patterns of the Swift{Hohenbergequation is explored when a continuous parameter n is varied that corresponds to the underlyingspace dimension whenever n is an integer. In particular, this numerical investigationreveals how 1D pulses and 2-pulses are connected to planar spots and rings when n isincreased from 1 to 2. It also elucidates changes in the snaking diagrams of spots whenthe dimension is switched from 2 to 3.A previously unknown spot solution is additionally uncovered. The second half of thethesis is devoted to rigorously proving this spot's existence. The amplitude of the spotexhibits an unexpected scaling as the bifurcation parameter is reduced to zero. The spotis constructed by gluing two known solutions together, each scaling as the square root ofthe bifurcation parameter, but it has a much larger scaling. This behaviour is explainedas a result of the proof.
Keywords
Pattern Formation
PDEs
Differential equations, Partial
Notes
Thesis (Ph.D. -- Brown University (2011)
Extent
xiii, 110 p.

Citation

McCalla, Scott G., "Localized Structures in the Multi-dimensional Swift-Hohenberg Equation" (2011). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:11254/

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