Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. L. Aero, Usp. Mekh., 1, No.3, 131 (2002).
M. Born and H. Kun, Dynamical Theory of Crystal Lattices (Intl. Ser. Monographs on Physics), Clarendon, Oxford (1954).
O. A. Ladyzhenskaya, Russ. Math. Surveys, 42, No.6, 27 (1987).
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, R. I. (1988).
J. K. Hale, L. T. Magalhaes, and W. M. Oliva, Dynamics in Infinite Dimensions, Springer, New York (2002).
A. B. Babin and M. I. Vishik, J. Math. Pures Appl., 62, 441 (1983); P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integrable Manifolds and Inertial Manifolds for Dissipative Differential Equations, Springer, New York (1989).
Yu. Il’yashenko and Weigu Li, Nonlocal Bifurcations [in Russian], MTsNMO CheRo, Moscow (1999); English transl., Amer. Math. Soc., Providence, R. I. (1999).
V. P. Maslov and G. A. Omel’yanov, Russ. Math. Surveys, 36, 73 (1981).
I. A. Molotkov and S. A. Vakulenko, Nonlinear Localized Waves [in Russian], Izdatel’stvo LGU, Leningrad (1988).
V. I. Arnol’d, V. S. Afrajmovich, Yu. S. Il’yashenko, and L. P. Shil’nikov, “Theory of bifurcations, ” in: Dynamical Systems 5 [in Russian] (Sovrem. Prob. Mat. Fund. Naprav., V. I. Arnol’d, ed.), VINITI, Moscow (1986), p. 5; English transl.: “Bifurcation theory,” in: Dynamical Systems V: Bifurcation Theory and Catastrophe Theory (Ency. Math. Sci., Vol. 5, V. I. Arnol’d, ed.), Springer, Berlin (1994), p. 7.
L. Simon, Ann. Math., 118, 525 (1983).
M. Jendoubi, J. Differential Equations, 144, 302 (1998).
P. Polacik, “Parabolic equations: asymptotic behavior and dynamics on invariant manifolds,” in: Handbook of Dynamical Systems (B. Fiedler, ed.), Vol. 2, Elsevier, Amsterdam (2002), p. 835.
D. Henry, J. Differential Equations, 59, 165 (1985).
T. Ohta and D. Jasnov, Phys. Rev. E, 56, 5648 (1997).
D. M. Petrich and R. E. Goldstein, Phys. Rev. Lett., 72, 1120 (1994).
S. A. Vakulenko, Ann. Inst. H. Poincare, 66, 373 (1997).
S. A. Vakulenko, Adv. Differential Equations, 5, 1739 (2000).
J. J. Hopfield, Proc. Nat. Acad. Sci. USA, 79, 2554 (1982).
M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore (1987).
R. Edwards, Math. Methods Appl. Sci., 19, 651 (1996).
Author information
Authors and Affiliations
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 357–367, June, 2005.
Rights and permissions
About this article
Cite this article
Aero, E.L., Vakulenko, S.A. Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice. Theor Math Phys 143, 782–791 (2005). https://doi.org/10.1007/s11232-005-0105-y
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11232-005-0105-y