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The Korteweg–de Vries, Burgers and Whitham limits for a spatially periodic Boussinesq model

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  30 August 2018

Roman Bauer ,
Wolf-Patrick Düll  and
Guido Schneider
Roman Bauer
Affiliation:
Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (roman.bauer@mathematik.uni-stuttgart.de; duell@mathematik.uni-stuttgart.de; guido.schneider@mathematik.uni-stuttgart.de)
Wolf-Patrick Düll
Affiliation:
Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (roman.bauer@mathematik.uni-stuttgart.de; duell@mathematik.uni-stuttgart.de; guido.schneider@mathematik.uni-stuttgart.de)
Guido Schneider
Affiliation:
Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (roman.bauer@mathematik.uni-stuttgart.de; duell@mathematik.uni-stuttgart.de; guido.schneider@mathematik.uni-stuttgart.de)
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Abstract

We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.

Keywords

spatially periodic Boussinesq modelKorteweg–de Vries (KdV) equationBurgers equationWhitham systemapproximation

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 1 , February 2019 , pp. 191 - 217
Copyright
Copyright © Royal Society of Edinburgh 2018 

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