Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves (\isolas"), or the length increases to infinity so that branches are unbounded in function space (\snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyze the structure of branches of localized radial roll solutions in dimension 1+", with 0 < " ϵ 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.
Bibliographical noteFunding Information:
∗Received by the editors October 3, 2018; accepted for publication (in revised form) May 28, 2019; published electronically August 1, 2019. https://doi.org/10.1137/18M1218728 Funding: The first author’s research was supported by an NSERC PDF. The second author’s research was supported by the NSF through grant DMS-1439786. The research of the third, fourth, and sixth authors was supported by the NSF through grant DMS-1148284. The seventh author’s research was partially supported by the NSF through grants DMS-1408742 and DMS-1714429. †Division of Applied Mathematics, Brown University, Providence, RI 02906 (jason firstname.lastname@example.org, Bjorn Sandstede@brown.edu). ‡Department of Mathematics, New York University, New York, NY 10012-1110 (email@example.com). §Department of Mathematics, University of Chicago, Chicago, IL 60637 (firstname.lastname@example.org). ¶Department of Mathematics, University of Texas at Austin, Austin, TX 78712 (email@example.com). ‖Department of Mathematics and Statistics, Boston University, Boston, MA 02215 (firstname.lastname@example.org). #Department of Mathematics, University of Arizona, Tucson, AZ 85721 (email@example.com).
© 2019 Society for Industrial and Applied Mathematics.
- Localized pattern
- Singular perturbation
- Snaking bifurcation
- Swift-Hohenberg equation