Commuting flows and conservation laws for noncommutative Lax hierarchies

Hamanaka, Masashi

doi:10.1063/1.1865321

Cited by 41 publications

(55 citation statements)

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“…There are several methods to construct such noncommutative extension of integrable systems. By deforming the Lax equation (for example, [21]) one can derive such equations. This is a bit ad hoc formalism and does not incorporate geometry.…”

Section: Motivation Results and Planmentioning

confidence: 99%

“…Once again, noncommutative deformation enormously enhances the spectrum of solitons. Several classical integrable models have been generalized to noncommutative spaces [10,21]. Also, under the Moyal deformation, the self-dual Yang-Mills equation is considered to preserve the integrability in the same sense as in commutative cases and contain new solutions special to noncommutative spaces [39].…”

Section: Nc Integrable Systems and Its Connection To Neighbouring Fieldsmentioning

confidence: 99%

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

Guha

2008

Acta Appl Math

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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 CamassaHolm 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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Section: Motivation Results and Planmentioning

confidence: 99%

Section: Nc Integrable Systems and Its Connection To Neighbouring Fieldsmentioning

confidence: 99%

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

Guha

2008

Acta Appl Math

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“…First we would like to comment on conservation laws of noncommutative field equations [23,16]. The discussion is basically the same as commutative case because both the differentiation and the integration are the same as commutative ones in the Moyal representation.…”

Section: Conservation Lawsmentioning

confidence: 99%

Noncommutative integrable systems and quasideterminants

Hamanaka

Xiu

Hu³

et al. 2010

AIP Conference Proceedings

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We discuss extension of soliton theories and integrable systems into noncommutative spaces. In the framework of noncommutative integrable hierarchy, we give infinite conserved quantities and exact soliton solutions for many noncommutative integrable equations, which are represented in terms of Strachan's products and quasi-determinants, respectively. We also present a relation to an noncommutative anti-self-dual Yang-Mills equation, and make comments on how "integrability" should be considered in noncommutative spaces.

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“…The ⋆− star product of two functions in (1+1)-dimensional noncommutative space is given by [7]- [9] (f ⋆ g) (t,…”

Section: Introductionmentioning

confidence: 99%