Local attractors of one of the original versions of the Kuramoto-Sivashinsky equation. (English. Russian original) Zbl 1519.35035
Theor. Math. Phys. 215, No. 3, 751-768 (2023); translation from Teor. Mat. Fiz. 215, No. 3, 339-359 (2023).
Summary: We study two rather similar evolutionary partial differential equations. One of them was obtained by Sivashinsky and the other by Kuramoto. The Kuramoto version was taken as the basic version of the equation that became known as the Kuramoto-Sivashinsky equation. We supplement each version of the Kuramoto-Sivashinsky equation with natural boundary conditions and, for the proposed boundary-value problems, study local bifurcations arising near a homogeneous equilibrium when they change stability. The analysis is based on the methods of the theory of dynamical systems with an infinite-dimensional phase space, namely, the methods of integral manifolds and normal forms. For all boundary-value problems, asymptotic formulas are obtained for solutions that form integral manifolds. We also point out boundary conditions under which the dynamics of solutions of the corresponding boundary-value problems of the two versions of the Kuramoto-Sivashinsky equation are significantly different.
MSC:
| 35B41 | Attractors |
| 35B32 | Bifurcations in context of PDEs |
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35K58 | Semilinear parabolic equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
Keywords:
Kuramoto-Sivashinsky equation; boundary-value problem; stability; bifurcation; invariant manifold; normal form; one space dimensionReferences:
| [1] | Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence (1984), Berlin: Springer, Berlin · Zbl 0558.76051 · doi:10.1007/978-3-642-69689-3 |
| [2] | Sivashinsky, G. I., Weak turbulence in periodic flow, Phys. D, 17, 243-255 (1985) · Zbl 0577.76045 · doi:10.1016/0167-2789(85)90009-0 |
| [3] | Godunov, S. K., Equations of Mathematical Physics (1979), Moscow: Nauka, Moscow |
| [4] | Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), New York: Springer, New York · Zbl 0871.35001 · doi:10.1007/978-1-4612-0645-3 |
| [5] | Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of the Kuramoto- Sivashinsky equations: nonlinear instability and attractors, Phys. D, 16, 155-183 (1985) · Zbl 0592.35013 · doi:10.1016/0167-2789(85)90056-9 |
| [6] | Armbruster, D.; Guckenheimer, J.; Holmes, P., Kuramoto-Sivashinsky dynamics on the center-unstable manifold, SIAM J. Appl. Math., 49, 676-691 (1989) · Zbl 0687.34036 · doi:10.1137/0149039 |
| [7] | Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Approximate inertial manifolds for the Kuramoto- Sivashinsky equation: analysis and computations, Phys. D, 44, 38-60 (1990) · Zbl 0704.58030 · doi:10.1016/0167-2789(90)90046-R |
| [8] | Kevrekidis, I. G.; Nicolaenko, B.; Scovel, J. C., Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50, 760-790 (1990) · Zbl 0722.35011 · doi:10.1137/0150045 |
| [9] | Larkin, N. A., Korteveg-de Vries and Kuramoto-Sivashinsky equations in bounded domain, J. Math. Anal. Appl., 297, 169-185 (2004) · Zbl 1075.35070 · doi:10.1016/j.jmaa.2004.04.053 |
| [10] | Bradley, R. M.; Harper, J. M. E., Theory of ripple topography induced by ion bombardment, J. Vac. Sci. Technol. A, 6, 2390-2395 (1988) · doi:10.1116/1.575561 |
| [11] | Emelyanov, V. I., The Kuramoto-Sivashinsky equation for the defect-deformation instability of a surface-stressed nanolayer, Laser Phys., 19, 538-543 (2009) · doi:10.1134/S1054660X0903030X |
| [12] | Emel’yanov, V. I., Defect-deformational surface layer instability as a universal mechanism for forming lattices and nanodot ensembles under the effect of ion and laser beams on solid bodies, Bull. Russ. Acad. Sci.: Phys., 74, 108-113 (2010) · Zbl 1236.82150 · doi:10.3103/S1062873810020024 |
| [13] | Kudryashov, N. A.; Ryabov, P. N.; Fedyanin, T. E., On self-organization processes of nanostructures on semiconductor surface by ion bombardment, Matem. Mod., 24, 23-28 (2012) · Zbl 1313.78017 |
| [14] | (ed.), V. I. Rudakov, Silicon Nanostructures. Physics. Technology. Modeling (2014), Yaroslavl: INDIGO, Yaroslavl |
| [15] | 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, 258, 11-46 (2013) · Zbl 1304.35184 · doi:10.1016/j.physd.2013.04.011 |
| [16] | Gelfand, M. P.; Bradley, R. M., One dimensional conservative surface dynamics with broken parity: arrested collapse versus coarsening, Phys. Lett. A, 379, 199-205 (2015) · doi:10.1016/j.physleta.2014.11.015 |
| [17] | Mikhlin, S. G., Mathematical Physics, An Advanced Course (1970), Amsterdam-London: North-Holland, Amsterdam-London · Zbl 0202.36901 |
| [18] | Naimark, M. A., Linear Differential Operators (1967), New York: Frederick Ungar, New York · Zbl 0219.34001 |
| [19] | Marsden, J. E.; McCracken, M., The Hopf Bifurcation and Its Applications (1976), New York: Springer, New York · Zbl 0346.58007 |
| [20] | Kulikov, A. N., Smooth invariant manifolds of a semigroup of nonlinear operators in a Banach space, Studies on Stability and the Theory of Oscillations, 114-129 (1976), Yaroslavl’: YarGU, Yaroslavl’ |
| [21] | Kulikov, A. N.; Kulikov, D. A., Formation of wavy nanostructures on the surface of flat substrates by ion bombardment, Comput. Math. Math. Phys., 52, 800-814 (2012) · Zbl 1274.82111 · doi:10.1134/S0965542512050132 |
| [22] | Kulikov, A. N.; Kulikov, D. A., Local bifurcations in the Cahn-Hilliard and Kuramoto-Sivashinsky equations and in their generalizations, Comput. Math. Math. Phys., 59, 630-643 (2019) · Zbl 1430.35021 · doi:10.1134/S0965542519040080 |
| [23] | Kulikov, A. N.; Kulikov, D. A., Cahn-Hilliard equation with two spatial variables. Pattern formation, Theoret. and Math. Phys., 207, 782-798 (2021) · Zbl 1471.35026 · doi:10.1134/S0040577921060088 |
| [24] | Kato, T., Perturbation Theory for Linear Operators (1995), Berlin: Springer, Berlin · Zbl 0836.47009 · doi:10.1007/978-3-642-66282-9 |
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