Well-posedness of modified Camassa-Holm equations
type: Item
instance of: scholarly article
title: Well-posedness of modified Camassa-Holm equations
zbMATH Open document ID: 1203.35079
author: Robert I. McLachlan
author: Xing-you Zhang
published in: Journal of Differential Equations
publication date: 2009-05-05T00:00:00Z
review text: This paper studies the Cauchy problem for the class of modified Camassa-Holm (mCH) equations given by \[ (\partial_t+u\partial_x+2u_x)m=0, \qquad m=(I-\partial_x^2)^ku, \tag{1} \] where \(k\geq2\) is a positive integer. The case \(k=1\) corresponds to the now celebrated Camassa-Holm (CH) equation. The authors begin by deriving Eq.~(1) as a geodesic equation on some diffeomorphism group with respect to the invariant \(H^k\) metric. Using Kato theory, they establish local well-posedness for the mCH equation and then proceed to prove global well-posedness (on the unit circle). Further well-posedness results are obtained for weak solutions of (1) among which is the Dirac delta function momentum (\(m=\delta\)) wave solution (that is analogous to the CH peakon wave). It is known that the CH equation has solutions which lead to blowup in finite time. Interestingly, this study shows that the solution dynamics for the mCH show significant differences to those for the CH equation. For example, it turns out that the mCH (1) does not admit finite-time blowup solutions for all \(k\geq2\). The study concludes with observations on the whole real-line case and some numerical simulations of (solitary-wave) solutions. This study will be of interest to those working on nonlinear wave equations -- especially shallow-water wave equations like the CH equation -- and the analytical and dynamical properties of their solutions.
reviewed by: A. Parker
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35C08
zbMATH DE Number: 5551136
zbMATH Keywords: modified Camassa-Holm equation
zbMATH Keywords: well-posedness
zbMATH Keywords: global existence
zbMATH Keywords: blowup
zbMATH Keywords: local well-posedness
zbMATH Keywords: global well-posedness
zbMATH Keywords: shallow-water wave equations
MaRDI profile type: Publication
full work available at URL: j.jde.2009.01.039
OpenAlex ID: W2040277751
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DOI: 10.1016/J.JDE.2009.01.039
description: scientific article; zbMATH DE number 5551136