G-Strands and Peakon Collisions on Diff(R)

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2013
• Hauptverfasser: Holm, D.D., Ivanov, R.I.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2013
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149231
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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.

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.