%0 Journal Article
%A Carpio Rodríguez, Ana María
%A Chapman, S. J.
%A Hastings, S.
%A Mcleod, J. B.
%T Wave solutions for a discrete reaction-diffusion equation
%D 2000
%@ 0956-7925
%U https://hdl.handle.net/20.500.14352/57218
%X Motivated by models from fracture mechanics and from biology, we study the infinite system of differential equations u'(n) = u(n-1) - 2u(n) + u(n+1) - A sin u(n) + F, ' = d/dt, where A and F are positive parameters. For fixed A > 0 we show that there are monotone travelling waves for F in an interval F-crit < F < A, and we are able to give a rigorous upper bound for F-crit, in contrast to previous work on similar problems. We raise the problem of characterizing those nonlinearities (apparently the more common) for which F-crit > 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\ < F-crit.
%~