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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, Volume 28, Issue 3, Pages 293–304
DOI: https://doi.org/10.20537/vm180303
(Mi vuu640)
 

This article is cited in 4 scientific papers (total in 4 papers)

MATHEMATICS

On the influence of the geometric characteristics of the region on nanorelief structure

D. A. Kulikov, A. V. Sekatskaya

Yaroslavl State University, ul. Sovetskaya, 14, Yaroslavl, 150003, Russia
Full-text PDF (237 kB) Citations (4)
References:
Abstract: The generalized Kuramoto–Sivashinsky equation in the case when the unknown function depends on two spatial variables is considered. This version of the equation is used as a mathematical model of formation of nonhomogeneous relief on a surface of semiconductors under ion beam. This equation is studied along with homogeneous Neumann boundary conditions in three regions: a rectangle, a square, and an isosceles triangle. The problem of local bifurcations in the case when spatially homogeneous equilibrium states change stability is studied. It is shown that for these three boundary value problems post-critical bifurcations occur and, as a result, spatially nonhomogeneous solutions bifurcate in each of these boundary value problems. For them asymptotic formulas are obtained. The dependence of the nature of bifurcations on the choice and geometry of the region is revealed. In particular, the type of dependence on spatial variables is determined. The problem of Lyapunov stability of spatially nonhomogeneous solutions is studied. Well-known methods from dynamical systems theory with an infinite-dimensional phase space: integral (invariant) manifolds, normal Poincare–Dulac forms in combination with asymptotic methods are used to analyze the bifurcation problems.
Keywords: Kuramoto–Sivashinsky equation, boundary-value problem, normal forms, stability, bifurcations.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00672_а
This work was supported by the Russian Foundation for Basic Research under Grant 18-01-00672.
Received: 19.03.2018
Bibliographic databases:
Document Type: Article
UDC: 517.956.4
MSC: 37H20
Language: Russian
Citation: D. A. Kulikov, A. V. Sekatskaya, “On the influence of the geometric characteristics of the region on nanorelief structure”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:3 (2018), 293–304
Citation in format AMSBIB
\Bibitem{KulSek18}
\by D.~A.~Kulikov, A.~V.~Sekatskaya
\paper On the influence of the geometric characteristics of the region on nanorelief structure
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 3
\pages 293--304
\mathnet{http://mi.mathnet.ru/vuu640}
\crossref{https://doi.org/10.20537/vm180303}
\elib{https://elibrary.ru/item.asp?id=35645983}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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    Abstract page:293
    Full-text PDF :179
    References:41
     
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