Preprint 90/2007
Virasoro Action on Pseudo-differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems
Partha Guha
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Submission date: 13. Sep. 2007
Pages: 23
MSC-Numbers: 17B68, 37K10, 58J40
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Abstract:
Using Grozman's formalism of invariant differential operators
we demonstrate the derivation of N = 2 Camassa-Holm equation
from the action of 
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
- 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 
on tensor densities of arbitrary weights.
Finally, using the formal Moyal deformed action of on the space of
Pseudo-differential symbols to derive the noncommutative analogues
of N = 1 super b- field equations.