Euler-poincaré formalism of coupled kdv type systems and diffeomorphism group on s 1

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This paper describes a wide class of coupled KdV equa- tions. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2- component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 met- ric on the semidirect product group Diff(S 1) × C (S 1), where Diff(S 1) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg- Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on Diff(S 1) × C (S 1).

Original languageBritish English
Pages (from-to)261-282
Number of pages22
JournalJournal of Applied Analysis
Issue number2
StatePublished - Dec 2005


  • Bott-virasoro group
  • Coupled KdV equations
  • Diffeomorphism
  • Geodesic flows
  • Twisted heisenberg-virasoro algebra


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