Geodesic
Invariant tori of the weakly dissipative version of the Ginzburg---Landau equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 216 (2022), pp. 66-75.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a periodic boundary value-problem for a weakly dissipative variant of the complex Ginzburg– Landau equation in the case where the period (wavelength) is small. The possibility of the existence of finite-dimensional invariant tori is proved. For solutions that belong to such tori, asymptotic formulas are obtained. We prove that all invariant tori, except for tori of dimension one (i.e., limit cycles), are unstable. We used various methods of the theory of dynamical systems with an infinite-dimensional space of initial conditions, for example, the method of integral (invariant) manifolds, the method of normal forms, and methods of perturbation theory.
Keywords: complex Ginzburg–Landau equation, periodic boundary-value problem, stability, asymptotic formula, small parameter.
Mots-clés : invariant torus 
@article{INTO_2022_216_a6,
     author = {A. N. Kulikov},
     title = {Invariant tori of the weakly dissipative version of the {Ginzburg---Landau} equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {66--75},
     publisher = {mathdoc},
     volume = {216},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_216_a6/}
}
TY  - JOUR
AU  - A. N. Kulikov
TI  - Invariant tori of the weakly dissipative version of the Ginzburg---Landau equation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 66
EP  - 75
VL  - 216
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_216_a6/
LA  - ru
ID  - INTO_2022_216_a6
ER  - 
%0 Journal Article
%A A. N. Kulikov
%T Invariant tori of the weakly dissipative version of the Ginzburg---Landau equation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 66-75
%V 216
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_216_a6/
%G ru
%F INTO_2022_216_a6
A. N. Kulikov. Invariant tori of the weakly dissipative version of the Ginzburg---Landau equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 216 (2022), pp. 66-75. http://geodesic.mathdoc.fr/item/INTO_2022_216_a6/

[14] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[15] Karyakin N. I., Bystrov K. N., Kireev P. S., Kratkii spravochnik po fizike, Vysshaya shkola, M., 1964

[16] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa tochechnykh otobrazhenii: printsip koltsa”, Differ. uravn., 39:5 (2003), 584–601 | MR

[17] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa tochechnykh otobrazhenii: sokhranenie invariantnogo tora pri vozmuscheniyakh”, Differ. uravn., 39:6 (2003), 738–-753 | MR

[18] Kolesov A. Yu., Kulikov A. N., Rozov N. X., “Tsilindricheskie beguschie volny obobschennogo kubicheskogo uravneniya Shredingera”, Dokl. RAN., 406:1 (2006), 21–29 | MR

[19] Kulikov A. N., “Inertsialnye invariantnye mnogoobraziya nelineinoi polugruppy v gilbertovom prostranstve”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 186 (2020), 57–66

[20] Landau L. D., Lifshits E. M., Kurs tteoreticheskoi fiziki. T. 6. Gidrodinamika, Nauka, M., 1988 | MR

[21] Lyusternik L. A., Sobolev V. I., Elementy funktsionalnogo analiza, GITTL, M.-L., 1951 | MR

[22] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950 | MR

[23] Yakubov S. Ya., “Razreshimost zadachi Koshi dlya abstraktnykh kvazilineinykh giperbolicheskikh uravnenii vtorogo poryadka i ikh prilozheniya”, Tr. Mosk. mat. o-va., 23 (1970), 37–60

[24] Aranson I. S., Kramer L., “The world of the complex Ginzburg–Landau equation”, Rev. Modern. Phys., 74 (2002), 99–143 | DOI | MR

[25] Bartuccelli M. V., Constantin P., Doering C. R., Gibbon J. D., Gisselfalt M., “On the possibility of soft and hard turbulence in the complex Ginzburg–Landau equation”, Phys. D., 44 (1990), 421–444 | DOI | MR

[26] Broer H. W., Dumortier F., van Strien S. J., Takens F., Structures in Dynamics: Finite Dimensional Deterministic Studies, 1991 | MR

[27] Kulikov A. N., Kulikov D. A., “Local bifurcations of plane running waves for the generalized cubic Schrödinger equation”, Differ. Equations., 46 (2010), 1299–1308 | DOI | MR

[28] Kuramoto Y., Tsusuki T., “On the formation of the dissipative structures in reaction-diffusion systems”, Progr. Teor. Phys., 54:3 (2018), 687–699 | DOI

[29] Scott A., Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford Univ. Press, London, 2003 | MR

[30] Segal I., “Nonlinear semigroups”, Ann. Math., 78:2 (1963), 339–364 | DOI | MR

[31] Witham G., Linear and Nonlinear Waves, Wiley, New York, 1974 | MR