## 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 H^{1} 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 language | British English |
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Pages (from-to) | 485-505 |

Number of pages | 21 |

Journal | Reviews in Mathematical Physics |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Jun 2010 |

## Keywords

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