Global weak solutions and breaking waves to the Degasperis-Procesi equation with linear dispersion
type: Item
instance of: scholarly article
title: Global weak solutions and breaking waves to the Degasperis-Procesi equation with linear dispersion
zbMATH Open document ID: 1180.35006
author: Fei Guo
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-06T00:00:00Z
review text: The author makes a new study of an initial value problem for the Degasperis-Procesi (DP) equation with strong linear dispersion in the form \[ u_{t}-u_{txx}+4u\, u_{x} + \gamma \left(u-u_{xx}\right)_{x} = 3u_{x}u_{xx}+u\, u_{xxx}, \] where \(\gamma (\in \mathbb{R})\) is a dispersive parameter. He then considers the both global strong and weak solutions of the above equation. This is followed by a study of precise breaking and permanent wave solution of the above equation. It is shown that, when the wave breaking occurs, there is at least one point where the slope of the wave profile is infinite at the breaking time. It is noted that the breaking point depends on the dispersive parameter \(\gamma\). Included is also the proof of existence and uniqueness of global weak solution of the above equation with strong dispersion. This is a new and interesting work on global weak solutions and breaking waves to the DP equation with linear dispersion.
reviewed by: Lokenath Debnath
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 5626911
zbMATH Keywords: Degasperis-Procesi equation
zbMATH Keywords: global weak solution
zbMATH Keywords: initial value problem
zbMATH Keywords: peaked solitons
zbMATH Keywords: strong dispersion
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.08.010
OpenAlex ID: W2025458968
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DOI: 10.1016/J.JMAA.2009.08.010
description: scientific article; zbMATH DE number 5626911