Geodesic
Second-kind equilibrium states of the Kuramoto--Sivashinsky equation with homogeneous Neumann boundary conditions
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 80-90.

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In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated.
Keywords: Kuramoto–Sivashinsky equation, boundary-value problem, equilibrium, stability, Galerkin method, computer analysis. 
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     author = {A. V. Sekatskaya},
     title = {Second-kind equilibrium states of the {Kuramoto--Sivashinsky} equation with homogeneous {Neumann} boundary conditions},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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A. V. Sekatskaya. Second-kind equilibrium states of the Kuramoto--Sivashinsky equation with homogeneous Neumann boundary conditions. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 80-90. http://geodesic.mathdoc.fr/item/INTO_2019_168_a9/

[1] Emelyanov V. M., “Defektno-deformatsionnaya neustoichivost kak universalnyi mekhanizm obrazovaniya reshetok i ansamblei nanotochek pri deistvii ionnykh i lazernykh puchkov na tverdye tela”, Izv. RAN. Ser. fiz., 2:74 (2010), 124–130 | Zbl

[2] Kudryashov N. A., Ryabov P. N., Strikhanov M. N., “Chislennoe modelirovanie formirovaniya nanostruktur na poverkhnosti ploskikh podlozhek pri ionnoi bombardirovke”, Yad. fiz. inzh., 2:1 (2010), 151–158

[3] Kudryashov N. A., Ryabov P. N., Fedyanin T. E., “Osobennosti samoorganizatsii nanostruktur na poverkhnosti poluprovodnikov pri ionnoi bombardirovke”, Mat. model., 12:24 (2012), 23–28 | Zbl

[4] Kulikov A. N., Kulikov D. A., “Formirovanie volnoobraznykh nanostruktur na poverkhnosti ploskikh podlozhek pri ionnoi bombardirovke”, Zh. vychisl. mat. mat. fiz., 2012, 930–945 | Zbl

[5] Kulikov A. N., Kulikov D. A., “Bifurkatsii prostranstvenno neodnorodnykh reshenii v dvukh kraevykh zadachakh dlya obobschennogo uravneniya Kuramoto—Sivashinskogo”, Vestn. MIFI., 3:4 (2014), 408–415 | DOI | MR

[6] Kulikov A. N., Kulikov D. A., “Uravnenie Kuramoto—Sivashinskogo. Lokalnyi attraktor, zapolnennyi neustoichivymi periodicheskimi resheniyami”, Model. anal. inform. sist., 1 (2018), 92–101

[7] Kulikov A. N., Kulikov D. A., Rudyi A. S., “Bifurkatsii nanostruktur pod vozdeistviem ionnoi bombardirovki”, Vestn. Udmurt. un-ta., 4 (2011), 86–99 | Zbl

[8] Sekatskaya A. V., “Bifurkatsii prostranstvenno neodnorodnykh reshenii v odnoi kraevoi zadache dlya obobschennogo uravneniya Kuramoto—Sivashinskogo”, Model. anal. inform. sist., 5:24 (2017), 615–628 | MR

[9] Sobolevskii P. E., “Ob uravneniyakh parabolicheskogo tipa v banakhovom prostranstve”, Tr. Mosk. mat. o-va., 1961, no. 10, 297–350

[10] Armbruster D., Guckenheimer J., Holmes Ph., “Kuramoto–Sivashinsky dynamics on the center-unstable manifold”, Siam J. Appl. Math., 3:49 (1989), 676–691 | DOI | MR | Zbl

[11] Barker B., Johnson M. A., Noble P., Zumbrun K., “Stability of periodic Kuramoto–Sivashinsky waves”, Appl. Math. Lett., 5:25 (2012), 824–829 | DOI | MR | Zbl

[12] Barker B., Johnson M. A., Noble P., Rodrigues L. M., Zumbrun K., “Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation”, Phys. D., 25 (2013), 11–46 | DOI | MR

[13] Bradley R., Harper J., “Theory of ripple topography induced by ion bombardment”, J. Vac. Sci. Technol., 4:6 (1988), 2390–2395 | DOI

[14] Emel'yanov B. I., “The Kuramoto–Sivashinsky equation for the defect-deformation. Instability of a surface-stressed nanolayer”, Laser Phys., 3:19 (2009), 538–543 | DOI

[15] Gelfand M. P., Bradley R. M., “One-dimensional conservative surface dynamics with broken parity: Arrested collapse versus coarsening”, Phys. Lett. A., 4:1 (2015), 199–205 | DOI | MR

[16] Kulikov A. N., Kulikov D. A., “Bifurcations in a boundary-value problem of nanoelectronics”, J. Math. Sci., 6 (2015), 211–221 | DOI | MR

[17] Kulikov A. N., Kulikov D. A., “Bifurcation in Kuramoto–Sivashinsky equation”, Pliska Stud. Math., 4:3 (2015), 101–110 | MR

[18] Kuramoto Y., Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984 | MR | Zbl

[19] Nicolaenko B., Scheurer B., Temam R., “Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors”, Phys. D., 12:24 (1985), 155–183 | DOI | MR

[20] Sivashinsky G. I., “Weak turbulence in periodic flow”, Phys. D., 2:17 (1985), 243–255 | DOI | MR