Erbay, Hüsnü AtaErbay, SaadetErkip, A.2016-12-062016-12-062016-111553-5231http://hdl.handle.net/10679/4568https://doi.org/10.3934/dcds.2016066In 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.enginfo:eu-repo/semantics/restrictedAccessArticle00038522130001010.3934/dcds.2016066Camassa-Holm equationImproved Boussinesq equationNonlocal wave equationRigorous justification2-s2.0-84984870611