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