Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2016) The CamassaHolm equation as the longwave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete and Continuous Dynamical Systems, 36 (11). pp. 61016116. ISSN 10780947 (Print) 15535231 (Online)
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Official URL: http://dx.doi.org/10.3934/dcds.2016066
Abstract
In the present study we prove rigorously that in the longwave 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 CamassaHolm 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 CamassaHolm 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 BenjaminBonaMahony approximation and the Kortewegde Vries approximation.
Item Type:  Article 

Uncontrolled Keywords:  CamassaHolm equation; improved Boussinesq equation; nonlocal wave equation; rigorous justification 
Subjects:  Q Science > QA Mathematics > QA299.6433 Analysis 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Albert Erkip 
Date Deposited:  03 Nov 2016 11:31 
Last Modified:  03 Nov 2016 11:31 
URI:  https://research.sabanciuniv.edu/id/eprint/29580 
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The CamassaHolm equation as the longwave limit of the improved Boussinsq equation and of a class of nonlocal wave equations. (deposited 25 Jan 2016 10:51)
 The CamassaHolm equation as the longwave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. (deposited 03 Nov 2016 11:31) [Currently Displayed]