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

H. A. Erbay; S. Erbay; A. Erkip

doi:10.3934/dcds.2017133

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2017, Volume 37, Issue 6: 3111-3122. Doi: 10.3934/dcds.2017133

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

1.
Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey
2.
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

* Corresponding author: H. A. Erbay
* Corresponding author: H. A. Erbay

Received: December 2016

Revised: January 2017

Published: May 2017

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Abstract

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional 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.

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Mathematics Subject Classification: Primary: 35Q53, 35Q74; Secondary: 74J30, 35C20.

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