  We consider a general class of convolution-type nonlocal wave equations
modeling bidirectional nonlinear wave propagation. The model involves two small
positive parameters measuring the relative strengths of the nonlinear and
dispersive effects. We take two different kernel functions that have similar
dispersive characteristics in the long-wave limit and compare the corresponding
solutions of the Cauchy problems with the same initial data. We prove
rigorously that the difference between the two solutions remains small over a
long time interval in a suitable Sobolev norm. In particular our results show
that, in the long-wave limit, solutions of such nonlocal equations can be well
approximated by those of improved Boussinesq-type equations.
