G-strands

Darryl D. Holm, Rossen I. Ivanov, James R. Percival

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

Abstract

A G-strand is a map g(t, s) : ℝ × ℝ→G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)- strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3) K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincaré system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) K-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)- strand is the G-strand version of the Sp(2) Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(ℝ)-strand equations on the diffeomorphism group G = Diff(ℝ) are also introduced and shown to admit solutions with singular support (e.g., peakons).

Original languageEnglish
Pages (from-to)517-551
Number of pages35
JournalJournal of Nonlinear Science
Volume22
Issue number4
DOIs
Publication statusPublished - Aug 2012

Keywords

  • Bloch-Iserles equation
  • Chiral models
  • Euler-Poincaré equations
  • Hamilton's principle
  • Integrable Hamiltonian systems
  • Inverse Spectral Transform (IST)
  • Momentum maps
  • Sobolev norms
  • Solitons
  • Spin chains

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