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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] Kulikov A. N., Kulikov D. A., “Formation of wavy nanostructures on the surface of flat substrates by ion bombardment”, Computational Mathematics and Mathematical Physics, 52:5 (2012), 800–814 | DOI | MR | Zbl

[2] Kulikov A. N., Kulikov D. A., Rudyi A. S., “Bifurcation of the nanostructures induced by ion bombardment”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, no. 4, 86–99 (in Russian) | DOI

[3] Sekatskaya A. V., “Bifurcations of spatially inhomogeneous solutions of a boundary value problem for the generalized Kuramoto–Sivashinsky equation”, Model. Anal. Inf. Sist., 24:5 (2017), 615–628 (in Russian) | DOI

[4] Bradley R. M., Harper J. M. E., “Theory of ripple topography induced by ion bombardment”, Journal of Vacuum Science and Technology A: Vacuum, Surfaces, and Films, 6:4 (1988), 2390–2395 | DOI

[5] Kudryashov N. A., Ryabov P. N., Strikhanov M. N., “Numerical modeling of the formation of nanostructures on the surface of flat substrates during ion bombardment”, Yadernaya Fizika i Inzhiniring, 1:2 (2010), 151–158 (in Russian)

[6] Functional analysis. Reference mathematical library, Nauka, M., 1972, 544 pp.

[7] Kuramoto Y., Chemical oscillations, waves and turbulence, Springer, Berlin, 1984, 156 pp. | DOI | MR | Zbl

[8] Sivashinsky G. I., “Weak turbulence in periodic flow”, Physica D: Nonlinear Phenomena, 17:2 (1985), 243–255 | DOI | MR | Zbl

[9] Nicolaenko B., Scheurer B., Temam R., “Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors”, Physica D: Nonlinear Phenomena, 16:2 (1985), 155–183 | DOI | MR | Zbl

[10] Foias C., Nicolaenko B., Sell G. R., Temam R., “Inertial manifolds for the Kuramoto–Sivashinsky equation and an estimate of their lowest dimension”, J. Math. Pures Appl. IX Ser., 67:3 (1988), 197–226 | MR | Zbl

[11] Kulikov A. N., Kulikov D. A., “Local bifurcations in the periodic boundary value problem for the generalized Kuramoto–Sivashinsky equation”, Automation and Remote Control, 78:11 (2017), 1955–1966 | DOI | MR | Zbl

[12] Kulikov A. N., Kulikov D. A., “Bifurcations of spatially nonhomogeneous solutions in two value boundary problems for generalized Kuramoto–Sivashinsky equation”, Vestnik Natsional'nogo Issledovatel'skogo Yadernogo Universiteta “MIFI”, 3:4 (2014), 408–415 (in Russian) | DOI | Zbl

[13] Kulikov A. N., Kulikov D. A., “Bifurcation in a boundary-value problem of nanoelectronics”, Journal of Mathematical Sciences, 208:2 (2015), 211–221 | DOI | MR | Zbl