Antiplectic structure, diffeomorphism and generalized KdV family

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Abstract

In this paper we show that the generalized KdV, generalized Camassa-Holm equations and the corresponding Möbius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by Vect(S1) on the space of differential operators and on the space of immersion curves, respectively. These are Euler-Poincaré type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler-Poincaré flow with respect to H1 metric, and this induces generalized Camassa-Holm equation. In the final section we discuss the Antiplectic pair in 2+1 dimensions.

Original languageBritish English
Pages (from-to)97-118
Number of pages22
JournalActa Applicandae Mathematicae
Volume91
Issue number2
DOIs
StatePublished - Apr 2006

Keywords

  • Antiplectic structure
  • Diffeomorphism
  • Generalized (Rudykh) KdV
  • Geodesic flows
  • KP family
  • Schwarzian CH
  • Schwarzian KdV

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