Euler-Poincaré flows on the loop Bott-Virasoro group and space of tensor densities and (2 + 1)-dimensional integrable systems

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Abstract

Following the work of Ovsienko and Roger ([54]), we study loop Virasoro algebra. Using this algebra, we formulate the Euler-Poincaré flows on the coadjoint orbit of loop Virasoro algebra. We show that the Calogero- Bogoyavlenskii-Schiff equation and various other (2 + 1)-dimensional KortewegdeVries (KdV) type systems follow from this construction. Using the right invariant H1 inner product on the Lie algebra of loop Bott-Virasoro group, we formulate the EulerPoincaré framework of the (2 + 1)-dimensional of the Camassa-Holm equation. This equation appears to be the Camassa-Holm analogue of the Calogero-Bogoyavlenskii-Schiff type (2 + 1)-dimensional KdV equation. We also derive the (2 + 1)-dimensional generalization of the Hunter-Saxton equation. Finally, we give an Euler-Poincaré formulation of one-parameter family of (1 + 1)-dimensional partial differential equations, known as the b-field equations. Later, we extend our construction to algebra of loop tensor densities to study the Euler-Poincaré framework of the (2 + 1)-dimensional extension of b-field equations.

Original languageBritish English
Pages (from-to)485-505
Number of pages21
JournalReviews in Mathematical Physics
Volume22
Issue number5
DOIs
StatePublished - Jun 2010

Keywords

  • (2 + 1)-dimensional Camassa equation
  • B-field equation
  • Calogero- Bogoyavlenskii-Schiff equation
  • Diffeomorphism
  • Loop Virasoro algebra
  • Tensor densities

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