Geodesic
Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 143 (2005) no. 3, pp. 357-367 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a system of hyperbolic nonlinear equations describing the dynamics of interaction between optical and acoustic modes of a complex crystal lattice (without a symmetry center) consisting of two sublattices. This system can be considered a nonlinear generalization of the well-known Born–Huang Kun model to the case of arbitrarily large sublattice displacements. For a suitable choice of parameters, the system reduces to the sine-Gordon equation or to the classical equations of elasticity theory. If we introduce physically natural dissipative forces into the system, then we can prove that a compact attractor exists and that trajectories converge to equilibrium solutions. In the one-dimensional case, we describe the structure of equilibrium solutions completely and obtain asymptotic solutions for the wave propagation. In the presence of inhomogeneous perturbations, this system is reducible to the well-known Hopfield model describing the attractor neural network and having complex behavior regimes.
Keywords: nonlinearity, attractor, complex behavior, neural networks. 
@article{TMF_2005_143_3_a2,
     author = {E. L. Aero and S. A. Vakulenko},
     title = {Asymptotic {Behavior} of {Solutions} of a {Strongly} {Nonlinear} {Model} of a {Crystal} {Lattice}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {357--367},
     year = {2005},
     volume = {143},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2005_143_3_a2/}
}
TY  - JOUR
AU  - E. L. Aero
AU  - S. A. Vakulenko
TI  - Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2005
SP  - 357
EP  - 367
VL  - 143
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2005_143_3_a2/
LA  - ru
ID  - TMF_2005_143_3_a2
ER  - 
%0 Journal Article
%A E. L. Aero
%A S. A. Vakulenko
%T Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2005
%P 357-367
%V 143
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2005_143_3_a2/
%G ru
%F TMF_2005_143_3_a2
E. L. Aero; S. A. Vakulenko. Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 143 (2005) no. 3, pp. 357-367. http://geodesic.mathdoc.fr/item/TMF_2005_143_3_a2/

[1] E. L. Aero, Uspekhi mekhaniki, 1:3 (2002), 131

[2] M. Born, Khuan Kun, Dinamicheskaya teoriya kristallicheskikh reshetok, IL, M., 1958

[3] O. A. Ladyzhenskaya, Uspekhi Mat. Nauk, 42:6 (1987), 25 | MR | Zbl

[4] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988 | MR | Zbl

[5] J. K. Hale, L. T. Magalhaes, W. M. Oliva, Dynamics in Infinite Dimensions, Spinger, N. Y., 2002 | MR

[6] A. B. Babin, M. I. Vishik, J. Math. Pures Appl., 62 (1983), 441 ; P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integrable Manifolds and Inertial Manifolds for Dissipative Differential Equations, Springer, New-York, 1989 | MR | Zbl | MR | Zbl

[7] Yu. Ilyashenko, Veigu Li, Nelokalnye bifurkatsii, MTsNMO CheRo, M., 1999 | MR

[8] V. P. Maslov, G. A. Omelyanov, Uspekhi mat. nauk, 36 (1981), 63 | Zbl

[9] I. A. Molotkov, S. A. Vakulenko, Nelineinye lokalizovannye volny, Izd-vo LGU, Leningrad, 1988 | MR

[10] V. I. Arnold, V. S. Afraimovich, Yu. S. Ilyashenko, L. P. Shilnikov, “Teoriya bifurkatsii”, Dinamicheskie sistemy – 5, Itogi nauki i tekhniki. Sovr. problemy mat. Fundamentalnye napravleniya, 5, ed. V. I. Arnold, VINITI, M., 1986, 5 | MR

[11] L. Simon, Annals. of Math., 118 (1983), 525 | DOI | MR | Zbl

[12] M. Jendoubi, J. Diff. Equ., 144 (1998), 302 | DOI | MR | Zbl

[13] P. Polácik, “Parabolic equations. Asymptotic behavior and dynamics on invariant manifolds”, Handbook of Dynamical Systems, V. 2, ed. B. Fiedler, Elsevier, Amsterdam, 2002, 835 | MR | Zbl

[14] D. Henry, J. Diff. Equ., 59 (1985), 165 | DOI | MR | Zbl

[15] T. Ohta, D. Jasnov, Phys. Rev. E, 56 (1997), 5648 | DOI

[16] D. M. Petrich, R. E. Goldstein, Phys. Rev. Lett., 72 (1994), 1120 | DOI | MR

[17] S. A. Vakulenko, Annales de L'Institut H. Poincarè Physique Théorique, 66 (1997), 373 | MR | Zbl

[18] S. A. Vakulenko, Adv. Diff. Equ., 5 (2000), 1739 | MR

[19] J. J. Hopfield, Proc. of Natl. Acad. USA, 79 (1982), 2554 | DOI | MR

[20] M. Mezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987 | MR | Zbl

[21] R. Edwards, Mathematical methods in the Applied Sciences, 19 (1996), 651 | 3.0.CO;2-S class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl