Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 027, 14 pages      arXiv:1211.6931      http://dx.doi.org/10.3842/SIGMA.2013.027
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

G-Strands 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 G-strand is a map g: R×RG 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.

Key words: Hamilton's principle; continuum spin chains; Euler-Poincaré equations; Sobolev norms; singular momentum maps; diffeomorphisms; harmonic maps.

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