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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| Дата: | 2013 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.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. |
Репозитарії
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irk-123456789-1492312019-02-20T01:23:15Z G-Strands and Peakon Collisions on Diff(R) Holm, D.D. Ivanov, R.I. 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. 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149231 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Holm, D.D. Ivanov, R.I. |
| spellingShingle |
Holm, D.D. Ivanov, R.I. G-Strands and Peakon Collisions on Diff(R) Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Holm, D.D. Ivanov, R.I. |
| author_sort |
Holm, D.D. |
| title |
G-Strands and Peakon Collisions on Diff(R) |
| title_short |
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_sort |
g-strands and peakon collisions on diff(r) |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149231 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:32:17Z |
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2023-05-20T17:32:17Z |
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1796153516190859264 |