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Russia Yaroslavl
Section  Mathematics 
Title  On the influence of the geometric characteristics of the region on nanorelief structure 
Author(s)  Kulikov D.A.^{a}, Sekatskaya A.V.^{a} 
Affiliations  Yaroslavl State University^{a} 
Abstract  The generalized KuramotoSivashinsky 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 postcritical 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. Wellknown methods from dynamical systems theory with an infinitedimensional phase space: integral (invariant) manifolds, normal PoincareDulac forms in combination with asymptotic methods are used to analyze the bifurcation problems. 
Keywords  KuramotoSivashinsky equation, boundaryvalue problem, normal forms, stability, bifurcations 
UDC  517.956.4 
MSC  37H20 
DOI  10.20537/vm180303 
Received  19 March 2018 
Language  Russian 
Citation  Kulikov D.A., Sekatskaya A.V. On the influence of the geometric characteristics of the region on nanorelief structure, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 293304. 
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