The Camassa-Holm equation as the long-wave limit of the improved   Boussinesq equation and of a class of nonlocal wave equations
  In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation. 
H. A. Erbay, S. Erbay and A. Erkip
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