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Tr. Mat. Inst. Steklova, 2008, Volume 261, Pages 188–209 (Mi tm748)  

This article is cited in 3 scientific papers (total in 3 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

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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2008, 261, 183–203

Bibliographic databases:

UDC: 517.958+517.91/.95+519.6
Received in October 2007

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, Tr. 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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\by N.~E.~Kulagin, L.~M.~Lerman, T.~G.~Shmakova
\paper On Radial Solutions of the Swift--Hohenberg Equation
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2008
\vol 261
\pages 188--209
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
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\pages 183--203
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    This publication is cited in the following articles:
    1. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer phenomenon in the spatially one-dimensional Swift–Hohenberg equation”, Proc. Steklov Inst. Math., 268 (2010), 130–147  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. N. E. Kulagin, L. M. Lerman, “Localized solutions of a piecewise linear model of the stationary Swift–Hohenberg equation on the line and on the plane”, Journal of Mathematical Sciences, 202:5 (2014), 684–702  mathnet  crossref
    3. van den Berg J.B., Groothedde C.M., Williams J.F., “Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg-Landau Problem”, SIAM J. Appl. Dyn. Syst., 14:1 (2015), 423–447  crossref  mathscinet  zmath  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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