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MiS Preprint
48/2008

Virasoro Action on Pseudo-differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Partha Guha

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.

Received:
Jul 28, 2008
Published:
Jul 29, 2008
MSC Codes:
17B68, 37K10, 58J40
Keywords:
pseudo-differential symbols, geodesic flow, noncommutative integrable systems

Related publications

inJournal
2009 Journal Open Access
Partha Guha

Virasoro action on pseudo-differential symbols and (noncommutative) supersymmetric peakon type integrable systems

In: Acta applicandae mathematicae, 108 (2009) 2, pp. 215-234