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In the present study we prove rigorously that in the long-wave limit, theunidirectional solutions of a class of nonlocal wave equations to which theimproved Boussinesq equation belongs are well approximated by the solutions ofthe Camassa-Holm equation over a long time scale. This general class ofnonlocal wave equations model bidirectional wave propagation in a nonlocallyand nonlinearly elastic medium whose constitutive equation is given by aconvolution integral. To justify the Camassa-Holm approximation we show thatapproximation errors remain small over a long time interval. To be moreprecise, we obtain error estimates in terms of two independent, small, positiveparameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity anddispersion, respectively. We further show that similar conclusions are alsovalid for the lower order approximations: the Benjamin-Bona-Mahonyapproximation and the Korteweg-de Vries approximation.

The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations