Abstract:
We consider the periodic boundary value problem for two variants of a weakly dissipative complex Ginzburg–Landau equation. In the first case, we study a variant of such an equation that contains the cubic and quintic nonlinear terms. We study the problem of local bifurcations of traveling periodic waves under stability changes. We show that a countable set of two-dimensional invariant tori arises as a result of such bifurcations. Both types of bifurcations are possible in the considered formulation of the problem, soft (postcritical) and hard (subcritical) ones, depending on the choice of the coefficients in the equation. We obtain asymptotic formulas for the solutions forming the invariant tori. We also study the periodic boundary value problem for the equation that is called the nonlocal Ginzburg–Landau equation in physics. We show that the boundary value problem in the considered variant has an infinite-dimensional global attractor. We present the solutions forming such an attractor.
Keywords:
Ginzburg–Landau equation, periodic boundary conditions, invariant manifold, single-mode solution, local bifurcation, global attractor, stability.
The work is performed in the framework of the
program for development of the regional mathematical center for
science and education (Yaroslavl State University) under financial
support from the Ministry of Science and Higher Education of Russian
Federation (Agreement on subsidy No. 075-02-2022-886 from Federal
budget).
Citation:
A. N. Kulikov, D. A. Kulikov, “Local bifurcations and a global attractor for two versions of the weakly dissipative Ginzburg–Landau equation”, TMF, 212:1 (2022), 40–61; Theoret. and Math. Phys., 212:1 (2022), 925–943
\Bibitem{KulKul22}
\by A.~N.~Kulikov, D.~A.~Kulikov
\paper Local bifurcations and a~global attractor for two versions of the~weakly dissipative Ginzburg--Landau equation
\jour TMF
\yr 2022
\vol 212
\issue 1
\pages 40--61
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\crossref{https://doi.org/10.4213/tmf10259}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4461543}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2022TMP...212..925K}
\transl
\jour Theoret. and Math. Phys.
\yr 2022
\vol 212
\issue 1
\pages 925--943
\crossref{https://doi.org/10.1134/S0040577922070042}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85134849638}
Linking options:
https://www.mathnet.ru/eng/tmf10259
https://doi.org/10.4213/tmf10259
https://www.mathnet.ru/eng/tmf/v212/i1/p40
This publication is cited in the following 2 articles:
D. A. Kulikov, “Mechanism for the formation of an inhomogeneous nanorelief and bifurcations in a nonlocal erosion equation”, Theoret. and Math. Phys., 220:1 (2024), 1122–1138
D. A. Kulikov, “Ustoichivost i lokalnye bifurkatsii odnomodovykh sostoyanii ravnovesiya variatsionnogo uravneniya Ginzburga–Landau”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:2 (2023), 240–258
Citation in format AMSBIB
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