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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Volume 261, Pages 188–209 (Mi tm748)  
This article is cited in 5 scientific papers (total in 5 papers)

On Radial Solutions of the Swift–Hohenberg Equation

N. E. Kulagina, L. M. Lermanb, T. G. Shmakovac

a State University of Management
b Research Institute for Applied Mathematics and Cybernetics, N. I. Lobachevski State University of Nizhnii Novgorod
c Moscow State Aviation Technological University
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Abstract: We study radial solutions to the generalized Swift–Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as $r\to\infty$. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift–Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo- and heteroclinic orbits, and numerical simulation.
Received in October 2007
English version:
Proceedings of the Steklov Institute of Mathematics, 2008, Volume 261, Pages 183–203
DOI: https://doi.org/10.1134/S0081543808020144
Bibliographic databases:
UDC: 517.958+517.91/.95+519.6
Language: Russian
Citation: N. E. Kulagin, L. M. Lerman, T. G. Shmakova, “On Radial Solutions of the Swift–Hohenberg Equation”, Differential equations and dynamical systems, Collected papers, Trudy Mat. Inst. Steklova, 261, MAIK Nauka/Interperiodica, Moscow, 2008, 188–209; Proc. Steklov Inst. Math., 261 (2008), 183–203
Citation in format AMSBIB
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\paper On Radial Solutions of the Swift--Hohenberg Equation
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
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\vol 261
\pages 188--209
\publ MAIK Nauka/Interperiodica
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    • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
     
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