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Zh. Vychisl. Mat. Mat. Fiz., 2019, Volume 59, Number 4, Pages 670–683 (Mi zvmmf10882)  

This article is cited in 2 scientific papers (total in 2 papers)

Local bifurcations in the Cahn–Hilliard and Kuramoto–Sivashinsky equations and in their generalizations

A. N. Kulikov, D. A. Kulikov

Yaroslavl State University, Yaroslavl, 150003 Russia

Abstract: A periodic boundary value problem for a nonlinear evolution equation that takes the form of such well-known equations of mathematical physics as the Cahn–Hilliard, Kuramoto–Sivashinsky, and Kawahara equations for specific values of its coefficients is studied. Three bifurcation problems arising when the stability of the spatially homogeneous equilibrium states changes are studied. The analysis of these problems is based on the method of invariant manifolds, the normal form techniques for dynamic systems with an infinite-dimensional space of initial conditions, and asymptotic methods of analysis. Asymptotic formulas for the bifurcation solutions are found, and stability of these solutions is analyzed. For the Kuramoto–Sivashinsky and Kawahara equations, it is proved that a two-dimensional local attractor exists such that all solutions on it are unstable in Lyapunov's sense.

Key words: nonlinear boundary value problem, stability, local bifurcations, normal form, asymptotic formulas.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00672_а
This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00672.


DOI: https://doi.org/10.1134/S0044466919040082


English version:
Computational Mathematics and Mathematical Physics, 2019, 59:4, 630–643

Bibliographic databases:

UDC: 517.958
Received: 08.11.2017
Revised: 14.11.2018
Accepted:14.11.2018

Citation: A. N. Kulikov, D. A. Kulikov, “Local bifurcations in the Cahn–Hilliard and Kuramoto–Sivashinsky equations and in their generalizations”, Zh. Vychisl. Mat. Mat. Fiz., 59:4 (2019), 670–683; Comput. Math. Math. Phys., 59:4 (2019), 630–643

Citation in format AMSBIB
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\paper Local bifurcations in the Cahn--Hilliard and Kuramoto--Sivashinsky equations and in their generalizations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2019
\vol 59
\issue 4
\pages 670--683
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\crossref{https://doi.org/10.1134/S0044466919040082}
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\transl
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\vol 59
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\pages 630--643
\crossref{https://doi.org/10.1134/S0965542519040080}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. N. Kulikov, D. A. Kulikov, “Cahn–Hilliard equation with two spatial variables. Pattern formation”, Theoret. and Math. Phys., 207:3 (2021), 782–798  mathnet  crossref  crossref  isi  elib
    2. A. N. Kulikov, D. A. Kulikov, “Attraktor obobschennogo uravneniya Kana—Khilliarda, vse resheniya na kotorom neustoichivy”, Materialy mezhdunarodnoi konferentsii po matematicheskomu modelirovaniyu v prikladnykh naukakh “International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19”. Belgorod, 20–24 avgusta 2019 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 195, VINITI RAN, M., 2021, 57–67  mathnet  crossref
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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