Localized radial roll patterns in higher space dimensions

Jason J. Bramburger; Dylan Altschuler; Chloe I. Avery; Tharathep Sangsawang; Margaret Beck; Paul Carter; Bjorn Sandstede

doi:10.1137/18M1218728

Localized radial roll patterns in higher space dimensions

Jason J. Bramburger, Dylan Altschuler, Chloe I. Avery, Tharathep Sangsawang, Margaret Beck, Paul Carter, Bjorn Sandstede

School of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	1420-1453
Number of pages	34
Journal	SIAM Journal on Applied Dynamical Systems
Volume	18
Issue number	3
DOIs	https://doi.org/10.1137/18M1218728
State	Published - 2019

Bibliographical note

Funding 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 [email protected], Bjorn [email protected]). ‡Department of Mathematics, New York University, New York, NY 10012-1110 ([email protected]). §Department of Mathematics, University of Chicago, Chicago, IL 60637 ([email protected]). ¶Department of Mathematics, University of Texas at Austin, Austin, TX 78712 ([email protected]). ‖Department of Mathematics and Statistics, Boston University, Boston, MA 02215 ([email protected]). #Department of Mathematics, University of Arizona, Tucson, AZ 85721 ([email protected]).

Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.

Keywords

Localized pattern
Singular perturbation
Snaking bifurcation
Swift-Hohenberg equation

Publisher link

10.1137/18M1218728

http://hdl.handle.net/10150/634921

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title = "Localized radial roll patterns in higher space dimensions",

abstract = "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.",

keywords = "Localized pattern, Singular perturbation, Snaking bifurcation, Swift-Hohenberg equation",

author = "Bramburger, {Jason J.} and Dylan Altschuler and Avery, {Chloe I.} and Tharathep Sangsawang and Margaret Beck and Paul Carter and Bjorn Sandstede",

note = "Funding 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{\textquoteright}s research was supported by an NSERC PDF. The second author{\textquoteright}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{\textquoteright}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 [email protected], Bjorn [email protected]). ‡Department of Mathematics, New York University, New York, NY 10012-1110 ([email protected]). §Department of Mathematics, University of Chicago, Chicago, IL 60637 ([email protected]). ¶Department of Mathematics, University of Texas at Austin, Austin, TX 78712 ([email protected]). ‖Department of Mathematics and Statistics, Boston University, Boston, MA 02215 ([email protected]). #Department of Mathematics, University of Arizona, Tucson, AZ 85721 ([email protected]). Publisher Copyright: {\textcopyright} 2019 Society for Industrial and Applied Mathematics.",

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AU - Altschuler, Dylan

AU - Avery, Chloe I.

AU - Sangsawang, Tharathep

AU - Beck, Margaret

AU - Carter, Paul

AU - Sandstede, Bjorn

N1 - Funding 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 [email protected], Bjorn [email protected]). ‡Department of Mathematics, New York University, New York, NY 10012-1110 ([email protected]). §Department of Mathematics, University of Chicago, Chicago, IL 60637 ([email protected]). ¶Department of Mathematics, University of Texas at Austin, Austin, TX 78712 ([email protected]). ‖Department of Mathematics and Statistics, Boston University, Boston, MA 02215 ([email protected]). #Department of Mathematics, University of Arizona, Tucson, AZ 85721 ([email protected]). Publisher Copyright: © 2019 Society for Industrial and Applied Mathematics.

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N2 - 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.

AB - 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.

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KW - Singular perturbation

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KW - Swift-Hohenberg equation

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Localized radial roll patterns in higher space dimensions

Abstract

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Keywords

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