We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
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.