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On the decoupling of the improved boussinesq equation into two uncoupled Camassa-Holm equations

Erbay, Hüsnü A. and Erbay, Saadet and Erkip, Albert (2017) On the decoupling of the improved boussinesq equation into two uncoupled Camassa-Holm equations. Discrete and Continuous Dynamical Systems, 37 (6). pp. 3111-3122. ISSN 1078-0947 (Print) 1553-5231 (Online)

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Official URL: http://dx.doi.org/10.3934/dcds.2017133

Abstract

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirecdional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and dispersion. Our results demonstrate that, in the present regime, any solution of the improved Boussinesq equation is split into two waves propagating in opposite directions independently, each of which is governed by the Camassa-Holm equation. We observe that the approximation error for the decoupled problem considered in the present study is greater than the approximation error for the unidirectional problem characterized by a single Camassa-Holm equation. We also consider lower order approximations and we state similar error estimates for both the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.

Item Type:Article
Uncontrolled Keywords:Camassa-Holm equation, improved Boussinesq equation, nonlocal wave equation, rigorous justification
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:31092
Deposited By:Albert Erkip
Deposited On:15 Mar 2017 15:51
Last Modified:15 May 2017 11:43

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