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

Partha Guha

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Submission date: 28. Jul. 2008
Pages: 23
published in: Acta applicandae mathematicae, 108 (2009) 2, p. 215-234 
DOI number (of the published article): 10.1007/s10440-008-9310-7
Bibtex
MSC-Numbers: 17B68, 37K10, 58J40
Keywords and phrases: pseudo-differential symbols, geodesic flow, noncommutative integrable systems
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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.