  In this work, we prove a comparison result for a general class of nonlinear
dispersive unidirectional wave equations. The dispersive nature of
one-dimensional waves occurs because of a convolution integral in space. For
two specific choices of the kernel function, the Benjamin-Bona-Mahony equation
and the Rosenau equation that are particularly suitable to model water waves
and elastic waves, respectively, are two members of the class. We first prove
an energy estimate for the Cauchy problem of the nonlocal unidirectional wave
equation. Then, for the same initial data, we consider two distinct solutions
corresponding to two different kernel functions. Our main result is that the
difference between the solutions remains small in a suitable Sobolev norm if
the two kernel functions have similar dispersive characteristics in the
long-wave limit. As a sample case of this comparison result, we provide the
approximations to the hyperbolic conservation law.
