The Frenkel-Kontorova Model: Concepts, Methods, and Applications (Theoretical and Mathematical Physics)

Theoretical physics deals with physical models. The main requirements for a good physical model are simplicity and universality. Universal models which can be applied to describe a variety of different phenomena are very rare in physics and, therefore, they are of key importance. Such models attract the special attention of researchers as they can be used to describe underlying physical concepts in a simple way. Such models appear again and again over the years and in various forms, thus extending their applicability and educa­ tional value. The simplest example of this kind is the model of a pendulum; this universal model serves as a paradigm which encompasses basic features of various physical systems, and appears in many problems of very different physical context. Solids are usually described by complex models with many degrees of freedom and, therefore, the corresponding microscopic equations are rather complicated. However, over the years a relatively simple model, known these days as the Prenkel-K ontorova model, has become one of the fundamental and universal tools of low-dimensional nonlinear physics; this model describes a chain of classical particles coupled to their neighbors and subjected to a pe­ riodic on-site potential.

This book presents an overview of the basic concepts, methods and applications of nonlinear low-dimensional solid-state physics based on the Frenkel--Kontorova model and its generalizations. It allows a nonspecialist to acquire foundations in the interdisciplinary concepts and methods of both solid-state physics and nonlinear science. It covers many important topics such as the nonlinear dynamics of discrete systems, the dynamics of solitons and their interaction, commensurate and incommensurate systems, statistical mechanics of nonlinear systems, ratchets, and nonequilibrium dynamics of interacting many-body systems. Also treated are the principal nonlinear equations along with the properties of their localized solutions and methods for their analysis.