Bifurcations of Spatially Inhomogeneous Solutions of a Boundary Value Problem for the Generalized Kuramoto–Syvashinsky Equation
https://doi.org/10.18255/1818-1015-2017-5-615-628
Abstract
In this paper, a differential partial equation with an unknown function of three variables time and two spatial variables – is considered. The given equation is commonly called the generalized Kuramoto–Sivashinsky (gKS) equation. This equation represents a model of the formation of a nanorelief on a surface by ion bombardment. In the work, this equation is considered with the homogeneous Neumann boundary conditions. Local bifurcations of spatially inhomogeneous equilibrium states is studied in the case of their stability changes. It is shown that the inhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. The conditions were obtained for coefficients when the stability changes. In the cases close to critical cases the local bifurcation problems are considered. It was shown that a question about the formation of inhomogeneous surface relief from a mathematical point of view is reduced to the study of auxiliary ordinary differential equations which are called a Poincare–Dulac normal form. The stability analysis of spatially homogeneous equilibrium states is given, as well as local bifurcations are studied in the case of their stability changes. The method of invariant manifolds coupled with the normal form theory were used to solve this problem. For the bifurcating solutions the asymptotic formulas are given.
About the Author
Alina V. Sekatskaya
P.G. Demidov Yaroslavl State University
Russian Federation
graduate student
Russian Federation
graduate student
References
1. Sigmund P., “A mechanism of surface micro-rougening by ion bombardment”, J. Mat. Sci., 8 (1973), 1545.
2. Sigmund P., “Theory of sputtering. Sputtering yield of amorphous and polycrystalline targets”, Phys. Rev, 184:2 (1969), 383–416.
3. Syvashinsky G. I., “Weak Turbulence in Periodic Flow”, Physica D: Nonlinear Phenomena, 17, 1985, 243–255.
4. Kuramoto Y., Chemical oscillations, waves and turbulence, Springer, Berlin, 1984, 136 pp.
5. Bradley R. M., Harper J. M. E., “Theory of Ripple Topography by ion bombardment”, J. Vac. Tech., A6, 1988, 2390–2395.
6. Kudryashov N. A., Ryabov P. N., Strikhanov M. N., “Chislennoe modelirovanie formirovaniya nanostruktur na poverkhnosti ploskikh podlozhek pri ionnoy bombardirovke”, Yadernaya fizika i inzhiniring, 1, 2, 2010, 151–158, (in Russian).]
7. Kudryashov N. A., Ryabov P. N., Fedyanin T. E., “On self-organization processes of nanostructures on semiconductor surface by ion bombardment”, Matem. Mod., 24:12 (2012), 23–28, (in Russian). ]
8. Carter G., “The physics and applications of ion beam erosion”, J.Phys. D.: Appl. Phys ˙ , 34, 2001.
9. Sigmund P., Sputtering by ion bombardment. Theoretical concepts. Sputtering by particle bombardment, Springer-Verlag, Berlin, 1981.
10. Funktsionalnyy analiz. Spravochnaya matematicheskaya biblioteka, Nauka, M., 1972, 544 pp., (in Russian).
11. Marsden Dzh., Mak-Kraken M., Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Mir, 1950, 367 pp., (in Russian).
12. Kulikov A. N., Integralnye mnogoobraziya nelineynykh avtonomnykh differentsialnykh uravneniy v gilbertovom prostranstve, 1991, (in Russian).
13. Mishchenko E. F., Sadovnichiy V. A., Kolesov A. Yu., Rozov N. Kh., Avtovolnovye protsessy v nelineynykh sredakh s diffuziey, Fizmatlit, 2005, 431 pp., (in Russian).
14. Lyusternik L. A., Sobolev V. I., Elementy funktsional’nogo analiza, Gos.izd-vo tekhniko-teoreticheskoy literatury, M.-L., 1951, 360 pp., (in Russian).
15. Kulikov A. N., Kulikov D. A., “Formation of wavy nanostructures on the surface of flat substrates by ion bombardment”, Comput. Math. and Math. Phys., 52:5 (2012), 800–814.
16. Kulikov A. N., Kulikov D. A., Rudyy A. S., “Bifurcation of the nanostructures induced by ion bombardment”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, 86–99, (in Russian). ]
17. Kulikov A. N., Kulikov D. A., “Bifurcations in a boundary-value problem of nanoelectronics”, Journal of Mathematical Sciences, 208:2 (2015), 211–221.
18. Kulikov A. N., Kulikov D. A., “Bifurcation in Kuramoto–Syvashinsky equation”, Pliska Stud. Math, 25 (2015), 81–90.
Review
For citations:
Sekatskaya A.V. Bifurcations of Spatially Inhomogeneous Solutions of a Boundary Value Problem for the Generalized Kuramoto–Syvashinsky Equation. Modeling and Analysis of Information Systems. 2017;24(5):615-628. (In Russ.) https://doi.org/10.18255/1818-1015-2017-5-615-628
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