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Symmetric solutions of evolutionary partial differential equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2017 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 10, 3932-3950 p.Article in journal (Refereed) Published
Abstract [en]

We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from EhrnstrOm et al (2009 Int. Math. Res. Not. 2009 4578-96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD , 2017. Vol. 30, no 10, 3932-3950 p.
Keyword [en]
evolution equations, symmetry, nonlocal equations, Euler equations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-215438DOI: 10.1088/1361-6544/aa8427ISI: 000411158000001Scopus ID: 2-s2.0-85030166337OAI: oai:DiVA.org:kth-215438DiVA: diva2:1150738
Note

QC 20171019

Available from: 2017-10-19 Created: 2017-10-19 Last updated: 2017-10-19Bibliographically approved

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Pei, Long

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • harvard1
  • 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