## Abstract

Using Grozman's formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S ^{1|2}) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H ^{1}-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S ^{1|1}) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S ^{1|1}) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.

Original language | British English |
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Pages (from-to) | 215-234 |

Number of pages | 20 |

Journal | Acta Applicandae Mathematicae |

Volume | 108 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2009 |

## Keywords

- Camassa-holm equation
- Geodesic flow
- Moyal deformation
- Noncommutative integrable systems
- Pseudo-differential symbols
- Super b-field equations
- Super KdV