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Contemporary Mathematics. Fundamental Directions, 2013, Volume 47, Pages 60–77 (Mi cmfd223)  
Localized solutions of a piecewise linear model of the stationary Swift–Hohenberg equation on the line and on the plane

N. E. Kulagina, L. M. Lermanbc

a State University of Management, Moscow, Russia
b Research Institute for Applied Mathematics and Cybernetics, Lobachevski Nizhnii Novgorod State University, Nizhnii Novgorod, Russia
c Lobachevski Nizhni Novgorod State University, Faculty of Mechanics and Mathematics, Nizhni Novgorod, Russia
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Abstract: In this paper we study a simplified model of the stationary Swift–Hohenberg equation, where the cubic nonlinearity is replaced by a piecewise linear function with similar properties. The main goal is to prove the existence of so-called localized solutions of this equation, i.e., solutions decaying to a homogeneous zero state with unbounded increase of the space variable. The following two cases of the space variable are considered: one-dimensional (on the whole line) and two-dimensional; in the latter case, radially symmetric solutions are studied. The existence of such solutions and increase of their number with change in the equation parameters are shown.
English version:
Journal of Mathematical Sciences, 2014, Volume 202, Issue 5, Pages 684–702
DOI: https://doi.org/10.1007/s10958-014-2072-z
Bibliographic databases:
Document Type: Article
UDC: 517.925.41+517.956.25
Language: Russian
Citation: 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”, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 3, CMFD, 47, PFUR, M., 2013, 60–77; Journal of Mathematical Sciences, 202:5 (2014), 684–702
Citation in format AMSBIB
\Bibitem{KulLer13}
\by N.~E.~Kulagin, L.~M.~Lerman
\paper Localized solutions of a~piecewise linear model of the stationary Swift--Hohenberg equation on the line and on the plane
\inbook Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2011). Part~3
\serial CMFD
\yr 2013
\vol 47
\pages 60--77
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd223}
\transl
\jour Journal of Mathematical Sciences
\yr 2014
\vol 202
\issue 5
\pages 684--702
\crossref{https://doi.org/10.1007/s10958-014-2072-z}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84921935983}
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