Geodesic
On the nature of local bifurcations of the Kuramoto--Sivashinsky equation in various domains
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 72-78.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a nonlinear parabolic partial differential equation in the case where the unknown function depends on two spatial variables and time, which is a generalization of the well-known Kuramoto–Sivashinsky equation. We consider homogeneous Dirichlet boundary-value problems for this equation. We examine local bifurcations when spatially homogeneous equilibrium states change stability. We show that post-critical bifurcations are realized in the boundary-value problems considered. We obtain asymptotic formulas for solutions and examine the stability of spatially inhomogeneous solutions.
Keywords: Kuramoto–Sivashinsky equation, boundary-value problem, equilibrium state, stability, Galerkin method, computer analysis. 
@article{INTO_2020_185_a7,
     author = {A. V. Sekatskaya},
     title = {On the nature of local bifurcations of the {Kuramoto--Sivashinsky} equation in various domains},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {72--78},
     publisher = {mathdoc},
     volume = {185},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_185_a7/}
}
TY  - JOUR
AU  - A. V. Sekatskaya
TI  - On the nature of local bifurcations of the Kuramoto--Sivashinsky equation in various domains
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 72
EP  - 78
VL  - 185
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_185_a7/
LA  - ru
ID  - INTO_2020_185_a7
ER  - 
%0 Journal Article
%A A. V. Sekatskaya
%T On the nature of local bifurcations of the Kuramoto--Sivashinsky equation in various domains
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 72-78
%V 185
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_185_a7/
%G ru
%F INTO_2020_185_a7
A. V. Sekatskaya. On the nature of local bifurcations of the Kuramoto--Sivashinsky equation in various domains. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M.T.Terekhin. Ryazan State University named for S.A. Yesenin, Ryazan, May 17-18, 2019. Part 1, Tome 185 (2020), pp. 72-78. http://geodesic.mathdoc.fr/item/INTO_2020_185_a7/

[4] Kremnievye nanostruktury. Fizika. Tekhnologiya. Modelirovanie, Izd-vo Indigo, Yaroslavl, 2014

[5] 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

[6] Kulikov A. N., “Attraktory dvukh kraevykh zadach dlya modifitsirovannogo nelineinogo telegrafnogo uravneniya”, Nelin. dinam., 4:1 (2008), 57–68 | MR

[7] Kulikov A. N., Kulikov D. A., “Formirovanie volnoobraznykh nanostruktur na poverkhnosti ploskikh podlozhek pri ionnoi bombardirovke”, Zh. vychisl. mat. mat. fiz., 52:5 (2012), 930–945 | MR | Zbl

[8] 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 | MR

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

[10] Kulikov A. N., Kulikov D. A., Rudyi A. S., “Bifurkatsii nanostruktur pod vozdeistviem ionnoi bombardirovki”, Vestn. Udmurt. un-ta. Mat. Mekh. Komp. nauki., 2011, no. 4, 86–99 | Zbl

[11] Kulikov D. A., Sekatskaya A. V., “O vliyanii geometricheskikh kharakteristik oblasti na strukturu nanorelefa”, Vestn. Udmurt. un-ta. Mat. Mekh. Komp. nauki., 28:3 (2018), 293–304 | MR | Zbl

[12] 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

[13] Funktsionalnyi analiz. Spravochnaya matematicheskaya biblioteka, Nauka, M., 1972

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

[15] 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

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

[17] 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

[18] 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

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

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

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

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

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

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