Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data...

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Main Authors: Zaihong Jiang, Sevdzhan Hakkaev
Format: Article
Language:English
Published: Hindawi Limited 2011-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2011/647368
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spelling doaj-715ae4c3183148418474e27be3a77a992020-11-24T21:40:13ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/647368647368Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water EquationsZaihong Jiang0Sevdzhan Hakkaev1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, ChinaFaculty of Mathematics and Informatics, Shumen University, 9712 Shumen, BulgariaWe investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solution u(t,x) with compactly supported initial datum u0(x) does not have compact x-support any longer in its lifespan.http://dx.doi.org/10.1155/2011/647368
collection DOAJ
language English
format Article
sources DOAJ
author Zaihong Jiang
Sevdzhan Hakkaev
spellingShingle Zaihong Jiang
Sevdzhan Hakkaev
Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
Abstract and Applied Analysis
author_facet Zaihong Jiang
Sevdzhan Hakkaev
author_sort Zaihong Jiang
title Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
title_short Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
title_full Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
title_fullStr Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
title_full_unstemmed Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations
title_sort wave breaking and propagation speed for a class of one-dimensional shallow water equations
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2011-01-01
description We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solution u(t,x) with compactly supported initial datum u0(x) does not have compact x-support any longer in its lifespan.
url http://dx.doi.org/10.1155/2011/647368
work_keys_str_mv AT zaihongjiang wavebreakingandpropagationspeedforaclassofonedimensionalshallowwaterequations
AT sevdzhanhakkaev wavebreakingandpropagationspeedforaclassofonedimensionalshallowwaterequations
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