## The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2016)
Official URL: http://dx.doi.org/10.3934/dcds.2016066 ## AbstractIn 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 is an element of 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.
## Available Versions of this Item- The Camassa-Holm equation as the long-wave limit of the improved Boussinsq equation and of a class of nonlocal wave equations. (deposited 25 Jan 2016 10:51)
- The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. (deposited 03 Nov 2016 11:31)
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- The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. (deposited 03 Nov 2016 11:31)
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