  We rigorously establish that, in the long-wave regime characterized by the
assumptions of long wavelength and small amplitude, bidirectional solutions of
the improved Boussinesq equation tend to associated solutions of two uncoupled
Camassa-Holm equations. We give a precise estimate for approximation errors in
terms of two small positive parameters measuring the effects of nonlinearity
and dispersion. Our results demonstrate that, in the present regime, any
solution of the improved Boussinesq equation is split into two waves
propagating in opposite directions independently, each of which is governed by
the Camassa-Holm equation. We observe that the approximation error for the
decoupled problem considered in the present study is greater than the
approximation error for the unidirectional problem characterized by a single
Camassa-Holm equation. We also consider lower order approximations and we state
similar error estimates for both the Benjamin-Bona-Mahony approximation and the
Korteweg-de Vries approximation.
