European Journal of Applied Mathematics



Wave solutions for a discrete reaction-diffusion equation


A. CARPIO a1, S. J. CHAPMAN a2, S. HASTINGS a3 and J. B. McLEOD a3
a1 Department of Applied Mathematics, Universidad Complutense de Madrid, Madrid, Spain
a2 Mathematical Institute , 24–29 St. Giles, Oxford OX1 3LB, UK
a3 Department of Mathematics, University of Pittsburgh, PA, USA

Abstract

Motivated by models from fracture mechanics and from biology, we study the infinite system of differential equations

formula here

where A and F are positive parameters. For fixed A > 0 we show that there are monotone travelling waves for F in an interval Fcrit < F < A, and we are able to give a rigorous upper bound for Fcrit, in contrast to previous work on similar problems. We raise the problem of characterizing those nonlinearities (apparently the more common) for which Fcrit > 0. We show that, for the sine nonlinearity, this is true if A > 2. (Our method yields better estimates than this, but does not include all A > 0.) We also consider the existence and multiplicity of time independent solutions when |F|< Fcrit.

(Received April 5 2000)