Generalized Variant of the Variational Ginzburg–Landau Equation
https://doi.org/10.1134/S2304487X20040045
Abstract
The version of the Ginzburg–Landau variational equation occurring in condensed matter physics which is called the ψ6-model is studied. As in the case of the ψ4-model, the corresponding equation is studied together with periodic boundary conditions. Conditions for the existence of single-mode equilibrium states are obtained for a periodic boundary value problem. An answer is given to the question of their stability in the sense of Lyapunov’s definition. In the case of a change in stability by single-mode equilibrium states, local bifurcations near these solutions are studied. It is shown that subcritical (rigid) bifurcations are characteristic of the problem under consideration. Bifurcation analysis involves the methods of the theory of dynamical systems with infinite-dimensional phase space and, first of all, the method of integral manifolds and the method of Poincaré normal forms. It is shown that two-dimensional invariant manifolds formed by spatially inhomogeneous solutions other than single-mode ones bifurcate from single-mode equilibrium states. For such solutions, asymptotic formulas are obtained.
Keywords
variational Ginzburg–Landau equation,
single-mode equilibrium states,
stability,
bifurcations,
spatial inhomogeneous solutions
Supporting agencies: The research was carried out with the financial support of the RFBR as part of a scientific project № 18-01-00672
About the Author
D. A. Kulikov
Yaroslavl State University named after P. G. Demidov
Russian Federation
Russian Federation
150003
Yaroslavl
References
1. Kuramoto Y. Chemical oscillations, waves and turbulence. Berlin: Springer-Verlag, 1984. 156 p.
2. Aronson I. S., Kramer L. The world of the complex Ginzburg-Landau equation // Reviews of modern physics. 2002. V. 74. P. 99–143.
3. Temam R. Infinite-dimensional dynamical systems in mechanics and physics. New-York: Springer-Verlag, 1997. 650 p.
4. Kulikov A. N., Rudy A. S. States of equilibrium of condensed matter within Ginzburg–Landau -model // Chaos, Solitons and Fractals. 2003. V. 15. P. 75–85.
5. Bartuccelli M., Constantin P., Doering Ch. R., Gibbon J. D., Gisselfalt M. On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation // Physica D. 1990. V. 44. P. 421–444.
6. Kulikov A. N., Kulikov D. A. Local bifurcations of plane running waves for the generalized cubic Schrodinger equation // Differential equations. 2010. V. 46. № 9. P. 1299–1308.
7. Malinetskiy G. G., Potapov A. B., Podlazov A. V. Nelineynaya dinamika. Podhody, resultaty, nadeshdy [Nonlinear dynamics. Approaches, results, hopes]. M.: Komkniga. 2006. 280 p. (in Russian).
8. Kulikov A. N., Kulikov D. A. Equilibrium states of the variational Ginzburg-Landau equation // Vestnik Natsional’nogo issledovatel’skogo yadernogo universiteta “MIFI”. 2017. V. 6. № 6. P. 496–502.
9. Krein S. G. Lineynie differentsialnye uravnenia v banachovom prostrantstve [Krein G. Linear differential equations in a Banach space]. M.: Nauka, 1977. 464 p. (in Russian).
10. Sobolevskiy P. E. Ob uravneniyach parabolicheskogo tipa v Banachovom prostrantstve [On equations of parabolic type in a Banach space]. Tr. MMO. 1967. V. 10. P. 297–350.
11. Sobolev S. L. Nekotorye primeneniya funkcionalnogo analiza v matematicheskoy fizike [Some applications of functional analysis in mathematical physics]. Leningrad. Isd-vo LGU. 1950. 256 p. (in Russian).
12. Guckenheimer J., Holmes P. J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Berlin.: Springer-Verlag. 1983. 462 p.
13. Kolesov A. Yu, Kulikov A. N., Rozov N. Kh. Invariant Tori of a class of point transformations: preservation of an invariant torus under perturbations // Differential equations. V. 39. № 6. P. 775–790.
14. Kulikov A. N., Kulikov D. A. Formation of wavy nanostructures on the surface of flat substrates by ion bombardment // Computational Mathematics and Mathematical Physics. 2012. V. 52. № 5. P. 800–814.
15. Kulikov A. N., Kulikov D. A. Local bifurcations in the Cahn-Hilliard and Kuramoto–Sivashinsky equations and in their generalizations // Computational Mathematics and Mathematical Physics. 2019. V. 59. № 4. P. 630–643.
Review
For citations:
Kulikov D.A. Generalized Variant of the Variational Ginzburg–Landau Equation. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(4):329-337. (In Russ.) https://doi.org/10.1134/S2304487X20040045
Views:
125
Email this article 