Derivation of the Camassa-Holm equations for elastic waves

In this paper we prove rigorously that in the long-wave limit the unidirectional solutions of the improved Boussinesq equation converge to the solutions of the associated Camassa-Holm equation. We frst provide a formal derivation of the Camassa-Holm equation from the improved Boussinesq equation using an asymptotic expansion valid as nonlinearity and dispersion parameters tend to zero independently. We then derive an evolution equation for a correction (remainder) term which represents the diference between solutions of the improved Boussinesq equation and the Camassa-Holm equation. Establishing suitable energy estimates for the remainder (uniformly in nonlinearity and dispersion parameters) we prove that, if the correction term is initially small, the correction term remains small for a relevant time interval. Keywords: Camassa-Holm equation, Improved Boussinesq equation, Asymptotic expansions, Rigorous justifcation.