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Existence and stability of spatially localized patterns.
- Source :
-
Journal of Differential Equations . Jan2019, Vol. 266 Issue 2/3, p1073-1120. 48p. - Publication Year :
- 2019
-
Abstract
- Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BIFURCATION theory
*SYMMETRY
*EIGENVALUES
*DIFFERENTIAL equations
*PLANAR sections
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 266
- Issue :
- 2/3
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 133092068
- Full Text :
- https://doi.org/10.1016/j.jde.2018.07.064