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.
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Guha, P. Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems. Acta Appl Math 108, 215–234 (2009). https://doi.org/10.1007/s10440-008-9310-7
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DOI: https://doi.org/10.1007/s10440-008-9310-7
Keywords
- Pseudo-differential symbols
- Super KdV
- Camassa-Holm equation
- Geodesic flow
- Super b-field equations
- Moyal deformation
- Noncommutative integrable systems