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

TL;DR

This paper rigorously proves that solutions of a broad class of nonlocal wave equations, modeling elastic media, are well approximated by the Camassa-Holm equation in the long-wave limit, with error estimates over extended time scales.

Contribution

It establishes the long-time validity of the Camassa-Holm equation as an approximation for nonlocal wave equations, including error bounds and applicability to lower order models.

Findings

Camassa-Holm equation approximates nonlocal wave solutions over long times

Error estimates depend on nonlinearity and dispersion parameters

Similar results hold for Benjamin-Bona-Mahony and Korteweg-de Vries models

Abstract

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.