Spatially Extended Monotone Mappings

Abstract

This chapter deals with the study of travelling waves in discrete time spatially extended systems with monotone dynamics. Such systems appear for instance in alloy solidification, in population dynamics and in solid-state physics. Special emphasis is made on the existence of travelling waves, on the uniqueness of their velocity and on their relevance for the description of propagation phenomena in such systems. The first section deals with interfaces between two stable homogeneous phases and their propagation in the form of fronts. The analysis applies to systems of bistable one-dimensional maps coupled via the convolution with an arbitrary distribution function [6]. This analysis completes a previous work on piecewise affine bistable CML [3, 4] and its extension to systems of piecewise affine one-dimensional maps coupled via convolution [5]. The second section deals with travelling waves in monotonous extended systems driven by spatially periodic forces. These systems are inspired by discrete time analogues of the dissipative dynamics of Frenkel-Kontorova models (see the chapters by Floŕia, Baesens and Gomez-Gardenes and by Baesens for such dynamics in continuous time). For such nonlinear systems, a dispersion relation is obtained and the existence of travelling waves with arbitrary wave and corresponding frequency is shown [7]. In spite of similarities with other works in the literature (see e.g. [12]) the methods and, particularly, the formalism developed in the papers [3, 4, 5, 6, 7] are quite distinct and original. They encompass in a unified framework, systems with continuous couplings and systems with discrete couplings. In particular, changes in the dynamics of travelling waves (e.g. changes in shape and in velocity) are described when the coupling continuously changes from a discrete to a continuous one.