Geodesic
On the influence of the geometric characteristics of the region on nanorelief structure
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 3, pp. 293-304

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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
Mots-clés : bifurcations. 
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     author = {D. A. Kulikov and A. V. Sekatskaya},
     title = {On the influence of the geometric characteristics of the region on nanorelief structure},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {293--304},
     publisher = {mathdoc},
     volume = {28},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2018_28_3_a2/}
}
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D. A. Kulikov; A. V. Sekatskaya. On the influence of the geometric characteristics of the region on nanorelief structure. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 3, pp. 293-304. http://geodesic.mathdoc.fr/item/VUU_2018_28_3_a2/