
SIGMA 9 (2013), 027, 14 pages arXiv:1211.6931
https://doi.org/10.3842/SIGMA.2013.027
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”
GStrands and Peakon Collisions on Diff(R)
Darryl D. Holm ^{a} and Rossen I. Ivanov ^{b}
^{a)} Department of Mathematics, Imperial College London, London SW7 2AZ, UK
^{b)} School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
Received October 29, 2012, in final form March 21, 2013; Published online March 26, 2013
Abstract
A Gstrand is a map g: R×R→G for a Lie group G that follows from Hamilton's
principle for a certain class of Ginvariant Lagrangians.
Some Gstrands on finitedimensional groups satisfy 1+1 spacetime 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 Gstrands for diffeomorphisms on the real line possess solutions with singular support
(e.g. peakons).
This paper studies collisions of such singular solutions of Gstrands 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 peakonantipeakon 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 Gstrand equations for G=Diff(R) corresponding to
a harmonic map g: C→Diff(R) and find explicit expressions for its peakonantipeakon
solutions, as well.
Key words:
Hamilton's principle; continuum spin chains; EulerPoincaré equations; Sobolev norms; singular momentum
maps; diffeomorphisms; harmonic maps.
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