Traveling Waves in Spatially Discrete Dynamical Systems of Diffusive Type

Abstract

We discuss recent results in the theory of lattice differential equations (LDE’s), particularly for LDE’s of diffusive type. Broadly speaking, an LDE is an infinite system of ordinary differential equations modeled on an underlying spatial lattice which reflects the underlying geometry of the lattice. Often one obtains an LDE upon discretizing a partial differential equation, however, many LDE’s occur as models in their own right and are not approximations to the continuum limit. Generally here we consider LDE’s composed of local nonlinear dynamics (often of bistable type) coupled with a discrete laplacian. Questions examined are the existence of equilibria (particularly those exhibiting regular patterns), the existence and qualitative properties of traveling waves, and the occurrence of pinning. We also describe in detail the necessary functional analysis associated with functional differential equations of mixed type, which play a central role in the proofs of these results.