(Noncommutative) Supersymmetric Peakon Type (original) (raw)
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. Keywords Pseudo-differential symbols • Super KdV • Camassa-Holm equation • Geodesic flow • Super b-field equations • Moyal deformation • Noncommutative integrable systems Mathematics Subject Classification (2000) 17B68 • 37K10 • 58J40 1 Prelude to Noncommutative Integrable Systems Noncommutative geometry [5] extends the notions of classical differential geometry from differential manifold to discrete spaces, like finite sets and fractals, and noncommutative spaces which are given by noncommutative associative algebras. It was an idea of Descartes that we can study a space by means of functions on the space, in other words, the algebra
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Acta Applicandae Mathematicae, 2008
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. Keywords Pseudo-differential symbols • Super KdV • Camassa-Holm equation • Geodesic flow • Super b-field equations • Moyal deformation • Noncommutative integrable systems Mathematics Subject Classification (2000) 17B68 • 37K10 • 58J40 1 Prelude to Noncommutative Integrable Systems Noncommutative geometry [5] extends the notions of classical differential geometry from differential manifold to discrete spaces, like finite sets and fractals, and noncommutative spaces which are given by noncommutative associative algebras. It was an idea of Descartes that we can study a space by means of functions on the space, in other words, the algebra
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After a comprehensive review of superconformal algebras, super-diffeomorphisms and supervector fields on supercircles S 1|n we study various supersymmetric extensions of the KdV and Camassa-Holm equations. We describe their (super) Hamiltonian structures and their connection to bihamiltonian geometry. These are interpreted as geodesic flows on various superconformal groups. We also give an example of superintegrable systems of Ramond type. The one-parameter family of equations shown by Degasperis, Holm and Hone (DHH) to possess multi-peakon solutions is identified as a geodesic flow equation on a one-parameter deformation of the group of diffeomorphisms of the circle, with respect to a right-invariant Sobolev H 1-metric. A supersymmetrisation of the algebra of deformed vector fields on S 1 yields supersymmetric DHH equations (also known as b-field equations), which include the supersymmetric Camassa-Holm equation as a special case.
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arXiv: High Energy Physics - Theory, 2000
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International Journal of Geometric Methods in Modern Physics, 2008
We use the logarithmic 2-cocycle and the action of V ect (S1) on the space of pseudodifferential symbols to derive one particular type of supersymmetric KdV equation, known as Kuper-KdV equation. This equation was formulated by Kupershmidt and it is different from the Manin–Radul–Mathieu type equation. The two Super KdV equations behave differently under a supersymmetric transformation and Kupershmidt version does not preserve SUSY transformation. In this paper we study the second type of supersymmetric generalization of the Camassa–Holm equation correspoding to Kuper-KdV equation via standard embedding of super vector fields into the Lie algebra of graded pseudodifferential symbols. The natural lift of the action of superconformal group SDiff yields SDiff module. This method is particularly useful to construct Moyal quantized systems.
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