G-Strands and Peakon Collisions on Diff(R)
A G-strand is a map g: R×R→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian syste...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2013 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2013
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149231 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | G-Strands and Peakon Collisions on Diff(R) / D.D. Holm, R.I. Ivanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862706953381216256 |
|---|---|
| author | Holm, D.D. Ivanov, R.I. |
| author_facet | Holm, D.D. Ivanov, R.I. |
| citation_txt | G-Strands and Peakon Collisions on Diff(R) / D.D. Holm, R.I. Ivanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A G-strand is a map g: R×R→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G=Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G=Diff(R) corresponding to a harmonic map g: C→Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well.
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| first_indexed | 2025-12-07T17:02:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-149231 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:02:48Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Holm, D.D. Ivanov, R.I. 2019-02-19T19:03:30Z 2019-02-19T19:03:30Z 2013 G-Strands and Peakon Collisions on Diff(R) / D.D. Holm, R.I. Ivanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J15; 37K05; 35R01 DOI: http://dx.doi.org/10.3842/SIGMA.2013.027 https://nasplib.isofts.kiev.ua/handle/123456789/149231 A G-strand is a map g: R×R→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G=Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G=Diff(R) corresponding to a harmonic map g: C→Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants
 and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html.
 We thank our friends A.M. Bloch, C.J. Cotter, F. Gay-Balmaz, A. Iserles, J.R. Percival,
 T.S. Ratiu and C. Tronci for their kind encouragement and thoughtful remarks during the
 course of this work. We are thankful also to Dr. Sergey Kushnarev and an anonymous referee
 whose comments and suggestions have helped us a lot in the revision of this paper. DDH gratefully acknowledges partial support by the Royal Society of London’s Wolfson Award scheme
 and the European Research Council’s Advanced Grant 267382 FCCA. RII is supported by the
 Science Foundation Ireland (SFI), under Grant No. 09/RFP/MTH2144. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications G-Strands and Peakon Collisions on Diff(R) Article published earlier |
| spellingShingle | G-Strands and Peakon Collisions on Diff(R) Holm, D.D. Ivanov, R.I. |
| title | G-Strands and Peakon Collisions on Diff(R) |
| title_full | G-Strands and Peakon Collisions on Diff(R) |
| title_fullStr | G-Strands and Peakon Collisions on Diff(R) |
| title_full_unstemmed | G-Strands and Peakon Collisions on Diff(R) |
| title_short | G-Strands and Peakon Collisions on Diff(R) |
| title_sort | g-strands and peakon collisions on diff(r) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149231 |
| work_keys_str_mv | AT holmdd gstrandsandpeakoncollisionsondiffr AT ivanovri gstrandsandpeakoncollisionsondiffr |