Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice

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.