Asymptotic research of local dynamics of the Cahn–Hilliard family equations
Cite this article as:
Plyshevskaya S. P. Asymptotic research of local dynamics of the Cahn–Hilliard family equations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 63-76. DOI: https://doi.org/10.18500/0869-6632-2019-27-1-63-76
Topic. Dynamics of well-known Cahn–Hilliard nonlinear equation is researched. In a state of balance stability task, critical cases were highlighted and bifurcation phenomena were researched. Aim. To formulate finite-dimensional and special infinite-dimensional equations, which can be represented as normal forms. Method. You can use as standard local dynamics research methods, based on constructing of normal forms on central manifolds, and special infinite-dimensional normalization ones. There is an algorithm of reducing an assumed boundary value task to equations for slowly varying amplitudes. Results. There are formulated finite-dimensional and special infinite-dimensional equations, which can be represented as normal forms. Their non-local dynamics defines the behavior of solutions that come from an assumed boundary value task minor adjacency. Asymptotic in between formulas to solve are quoted as well. Discussion. An offered problem is divided into a continual family, which depends on a certain parameter of more specialized boundary value tasks. As a rule, considered critical cases possess 1 and 2 dimensions. You’ve got a situation that is inherent to advection index major values, when a critical case possesses an infinite advection: infinitely many roots of a characteristic equation of a linearized boundary value problem aim for an imaginary axis with this index increase.
1. Krasnyuk I.B., Stefanovic L.I., Yurchenko V.M. Analysis of oscillations of concentration in bounded binary mixtures taking into account surface effects. Technical Physics, 2007, vol. 52, no. 11, pp. 1445–1452.
2. Cahn J.W., Hilliard J.E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phis., 1958, vol. 28, pp. 258–267.
3. Kashchenko S.A. Bifurcations in Kuramoto–Sivashinsky equations. Theoretical and Mathematical Physics, 2017, vol. 192, no. 1, pp. 958–973.
4. Marsden J.E., McCracken M. The Hopf Bifurcation and its Applications. Berlin: Springer, Applied Math. Series, no. 19, 1976, 408 p.
5. Bruno A.D. Local Method of Nonlinear Analysis of Differential Equations. Moscow: Nauka, 1979 (in Russian).
6. Kashchenko S.A., Preobrazhenskaya M.M. Bifurcations in the generalized Korteweg–de Vries equation. Russian Mathematics, 2018, vol. 62, no. 2, pp. 49–61.
7. Butuzov V.F., Vasil’yeva A.B., Nefodov N.N. Asymptotic theory of contrasting structures. A survey. Automation and Remote Control, 1997, vol. 58, no. 7, 1068–1091.
8. Butuzov V.F., Levashova N.T. On a system of reaction–diffusion–transfer type in the case of small diffusion and fast reactions. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 7, pp. 962–974.
9. Kashchenko S.A. Normal form for the Korteweg–de Vries–Burgers equation. Doklady Akademii Nauk, 2016, vol. 468, no. 4, pp. 383–386.
BibTeX
author = {S. P. Plyshevskaya},
title = {Asymptotic research of local dynamics of the Cahn–Hilliard family equations},
year = {2019},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {27},number = {1},
url = {https://old-andjournal.sgu.ru/en/articles/asymptotic-research-of-local-dynamics-of-the-cahn-hilliard-family-equations},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2019-27-1-63-76},pages = {63--76},issn = {0869-6632},
keywords = {dynamics,stability,normal forms,the Cahn–Hilliard equation},
abstract = {Topic. Dynamics of well-known Cahn–Hilliard nonlinear equation is researched. In a state of balance stability task, critical cases were highlighted and bifurcation phenomena were researched. Aim. To formulate finite-dimensional and special infinite-dimensional equations, which can be represented as normal forms. Method. You can use as standard local dynamics research methods, based on constructing of normal forms on central manifolds, and special infinite-dimensional normalization ones. There is an algorithm of reducing an assumed boundary value task to equations for slowly varying amplitudes. Results. There are formulated finite-dimensional and special infinite-dimensional equations, which can be represented as normal forms. Their non-local dynamics defines the behavior of solutions that come from an assumed boundary value task minor adjacency. Asymptotic in between formulas to solve are quoted as well. Discussion. An offered problem is divided into a continual family, which depends on a certain parameter of more specialized boundary value tasks. As a rule, considered critical cases possess 1 and 2 dimensions. You’ve got a situation that is inherent to advection index major values, when a critical case possesses an infinite advection: infinitely many roots of a characteristic equation of a linearized boundary value problem aim for an imaginary axis with this index increase. }}