G-Strands and Peakon Collisions on Diff(R)
Vernadsky National Library of Ukraine
Переглянути архів Інформація| Поле | Співвідношення | |
| Title |
G-Strands and Peakon Collisions on Diff(R)
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| Creator |
Holm, D.D.
Ivanov, R.I. |
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| 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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| Date |
2019-02-19T19:03:30Z
2019-02-19T19:03:30Z 2013 |
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| Type |
Article
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| Identifier |
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 |
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| Language |
en
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| Relation |
Symmetry, Integrability and Geometry: Methods and Applications
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| Publisher |
Інститут математики НАН України
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