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G-Strands and Peakon Collisions on Diff(R)
- Source :
- Symmetry, Integrability and Geometry: Methods and Applications, Vol 9, p 027 (2013)
- Publication Year :
- 2013
- Publisher :
- National Academy of Science of Ukraine, 2013.
-
Abstract
- 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.
Details
- Language :
- English
- ISSN :
- 18150659
- Volume :
- 9
- Database :
- Directory of Open Access Journals
- Journal :
- Symmetry, Integrability and Geometry: Methods and Applications
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.bd9d440933ad484b90cd0ba88f0a0e6d
- Document Type :
- article
- Full Text :
- https://doi.org/10.3842/SIGMA.2013.027