Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-4137-8272
Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada.
2017 (English)In: Journal of Integrable Systems, ISSN 2058-5985, Vol. 2, no 1, xyw014Article in journal (Refereed) Published
Abstract [en]

We consider multipeakon solutions, and to some extent also multishockpeakon solutions, of a coupled two- component integrable PDE found by Geng and Xue as a generalization of Novikov’s cubically nonlinear Camassa–Holm type equation. In order to make sense of such solutions, we find it necessary to assume that there are no overlaps, meaning that a peakon or shockpeakon in one component is not allowed to occupy the same position as a peakon or shockpeakon in the other component. Therefore one can distinguish many inequivalent configurations, depending on the order in which the peakons or shockpeakons in the two components appear relative to each other. Here we are particularly interested in the case of interlacing peakon solutions, where the peakons alternatingly occur in one component and in the other. Based on explicit expressions for these solutions in terms of elementary functions, we describe the general features of the dynamics, and in particular the asymptotic large-time behaviour (assuming that there are no antipeakons, so that the solutions are globally defined). As far as the positions are concerned, interlacing Geng–Xue peakons display the usual scattering phenomenon where the peakons asymptotically travel with constant velocities, which are all distinct, except that the two fastest peakons (the fastest one in each component) will have the same velocity. However, in contrast to many other peakon equations, the amplitudes of the peakons will not in general tend to constant values; instead they grow or decay exponentially. Thus the logarithms of the amplitudes (as functions of time) will asymptotically behave like straight lines, and comparing these lines for large positive and negative times, one observes phase shifts similar to those seen for the positions of the peakons (and also for the positions of solitons in many other contexts). In addition to these K+K interlacing pure peakon solutions, we also investigate 1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2 peakon–antipeakon solution.

Place, publisher, year, edition, pages
Oxford University Press, 2017. Vol. 2, no 1, xyw014
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-139590DOI: 10.1093/integr/xyw014OAI: oai:DiVA.org:liu-139590DiVA: diva2:1130351
Available from: 2017-08-09 Created: 2017-08-09 Last updated: 2017-08-30Bibliographically approved

Open Access in DiVA

fulltext(663 kB)6 downloads
File information
File name FULLTEXT01.pdfFile size 663 kBChecksum SHA-512
d01aa27be1e321bd92d6465f6a855114725de9ef32f978e09871e8706c6bc814f949fcbb0f3d1f8dac5ac374adec7167a5b658f064173c86dbdf6b599cd06ccd
Type fulltextMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Lundmark, Hans
By organisation
Mathematics and Applied MathematicsFaculty of Science & Engineering
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 6 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 86 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf