Soliton In One-dimensional Nonlinear Lattices

Soliton is a universal and important nonlinear phenomenon. Most of the earlier works focused on the soliton in continuum media. In recent years, however, more and more attention has been paid to the soliton in the discrete nonlinear lattices. In 1988, Sievers et al firstly showed that there exist intrinsic localized modes (ILMs) in the perfect lattices resulting from the interplay between discreteness and nonlinearity. Although their properties are analogous to the impurity-induced localized modes in the linear lattices they can lie on any site of the lattices. Henceforth, solitons in the nonlinear lattice systems became an active research realm. The one-dimensional (1D) chain of atoms is the simplest lattice model. Because of the simplicity of the model, its involvement of the transparent physical significance and support for various kinds of nonlinear excitations it attracted considerable interests. In the study of the parametrically driven and damped chain of coupled pendulum, Denardo et al demonstrated experimentally that when the dissipation is balanced by the driven, parametrically driven standing waves emerge in 1990. Hereafter, a vast number of theoretical and experimental studies were presented by researchers at home and abroad. We also should note that studies of the interaction between solitons and impurities are frequently reported in recent years. One of the most remarkable findings is that by introducing a single impurity into an array of parametrically driven nonlinear oscillators, the impurity can pin stationary wave and tame spatiotemporal chaos. However, the analytical study in this direction is rather rare. In this paper, we study the solitons in 1D atomic chain and nonlinear coupled chain of pendulum analytically by means of the method mainly based on multiple-scale technique, some significant results are presented. These results not only enrich the soliton theory in the 1D system but also provide some theoretical basis for other theoretical and experimental works of relevancy.The thesis consists of four chapters: In chapter one, we briefly introduce the basic theoretical concept of soliton, the history, present status and significance of the study of solitons. Chapter two is devoted to depicting the basic principle and handling technique of 1D discrete lattice systems, particular attention is paid to introducing some widely used analytical approaches and related experimental technique. In Chapter three, we develop a new analytical method in the study of the lattice dynamics of the diatomic chain at the boundary of the Brillouin zone (BZ), a...