Spatially localized structures, in which a spatially oscillatory pattern on a finite spatial range connects to a trivial homogeneous solution outside this range, have been observed in numerous physical contexts, including cellular buckling, plane Couette flow, vegetation patterns, optical cavity solitons, crime hotspots, and many others. Despite the widely disparate contexts in which they arise, the bifurcation diagrams of such patterns often exhibit similar snaking behavior, in which branches of symmetric solutions, connected by bifurcating branches of asymmetric solutions, wind back and forth between two limits of an appropriate parameter. In this thesis we address the existence and stability of stationary localized solutions of parabolic partial differential equations (PDEs) on the line and the plane. One particular model system supporting localized structures is the Swift--Hohenberg system, and we use this system for numerical illustration of our existence and stability results.
Our main results are as follows: we give a new proof of the existence asymmetric localized structures, utilizing information about the underlying front structure and providing a unified approach to the existence of all localized structures. This enables a rigorous proof of the stability properties of symmetric and asymmetric structures. We show that the temporal eigenvalues of localized structures in the right half plane are exponentially close to those of the front and back added with multiplicity, and furthermore that the eigenvalue at the origin remains simple. We then address numerical results showing unexpected behavior of eigenvalues within the essential (or absolute) spectrum, and propose an analytical explanation of these results. We conclude by predicting the results of perturbative terms in PDE systems supporting localized snaking solutions, and make qualitative and quantitative predictions for topological changes to the associated bifurcation diagrams, as well as drift speeds of particular solutions.
"Existence and Stability of Spatially Localized Planar Patterns"
Applied Mathematics Theses and Dissertations.
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