Abstract
A boundary-value problem for the generalized Kuramoto–Sivashinsky equation with homogeneous Neumann boundary conditions is considered. The stability of spatially homogeneous equilibrium states are analyzed and local bifurcations at change of stability are studied. We use the method of invariant manifolds in combination with the theory of normal forms. Asymptotic formulas for bifurcating solutions are found.
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References
R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A, 6, No. 4, 2390–2395 (1988).
C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam, “Inertial manifolds for the Kuramoto–Sivashinsky equation and an estimate of their lowest dimension,” J. Math. Pures Appl., 67, 197–226 (1988).
C. Foias and E. C. Titi, “Determining nodes, finite difference schemes and inertial manifolds,” Nonlinearity, 4, No. 1, 135–153 (1991).
S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1977).
A. N. Kulikov, “On smooth invariant manifolds of the semigroup of nonlinear operators in a Banach space,” in: Studies on Stability and Theory of Oscillations [in Russian], Yaroslavl (1976), pp. 114–129.
A. N. Kulikov and D. A. Kulikov, “Formation of wave-like nanostructures on the surface of flat substrates during ion bombardment,” Zh. Vychisl. Mat. Mat. Fiz., 52, No. 5, 930–945 (2012).
A. N. Kulikov and D. A. Kulikov, “Bifurcations in a boundary-value problem of nanoelectronics,” J. Math. Sci., 208, No. 2, 211–221 (2015).
A. N. Kulikov and D. A. Kulikov, “Bifurcations in Kuramoto–Sivashinsky equation,” Pliska Stud. Math., 25, 101–110 (2015).
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin (1984).
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Appl. Math. Sci., 19, Springer-Verlag (1976).
B. Nicolaenko, B. Scheurer, and R. Temam, “Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors,” Physica D, 16, No. 2, 155–183 (1985).
I. Sivashinsky, “Weak turbulence in periodic flow,” Phys. D, 17, No. 2, 243–255 (1985).
P. E. Sobolevsky, “On parabolic equations in Banach spaces,” Tr. Mosk. Mat. Obshch., 10, 297–350 (1967).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 148, Proceedings of the International Conference “Geometric Methods in Control Theory and Mathematical Physics: Differential Equations, Integrability, and Qualitative Theory” (Ryazan, September 15–18, 2016), 2018.
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Kulikov, A.N., Sekatskaya, A.V. Local Attractors in a Certain Boundary-Value Problem for the Kuramoto–Sivashinsky Equation. J Math Sci 248, 430–437 (2020). https://doi.org/10.1007/s10958-020-04883-1
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DOI: https://doi.org/10.1007/s10958-020-04883-1