Extensions of Soliton equations to non-commutative (2 + 1) dimensions

Toda, Kouichi

doi:10.22323/1.008.0038

Cited by 13 publications

(30 citation statements)

References 16 publications

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“…Supersymmetric extension (e.g 114,133 ) and higher dimensional extension (e.g. 162 ) would be interesting and straightforwardly possible. Extension to non(-anti)commutative superspaces is also considerable.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Noncommutative Solitons and Integrable Systems

Hamanaka

2005

Noncommutative Geometry and Physics

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We review recent developments of soliton theories and integrable systems on noncommutative spaces. The former part is a brief review of noncommutative gauge theories focusing on ADHM construction of noncommutative instantons. The latter part is a report on recent results of existence of infinite conserved densities and exact multi-soliton solutions for noncommutative Gelfand-Dickey hierarchies. Some examples of noncommutative Ward's conjecture are also presented. Finally, we discuss future directions on noncommutative Sato's theories and twistor theories.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Noncommutative Solitons and Integrable Systems

Hamanaka

2005

Noncommutative Geometry and Physics

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“…NC version of it was first derived in [42]. Here, we derive the equation from NC ASDYM equation by reduction as follows.…”

Section: Reduction To Nc Cbs Equationmentioning

confidence: 99%

Non-commutative Ward's conjecture and integrable systems

Hamanaka

2006

Nuclear Physics B

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Noncommutative Ward's conjecture is a noncommutative version of the original Ward's conjecture which says that almost all integrable equations can be obtained from anti-selfdual Yang-Mills equations by reduction. In this paper, we prove that wide class of noncommutative integrable equations in both (2 + 1)-and (1 + 1)-dimensions are actually reductions of noncommutative anti-self-dual Yang-Mills equations with finite gauge groups, which include noncommutative versions of Calogero-Bogoyavlenskii-Schiff eq., Zakharov system, Ward's chiral and topological chiral models, (modified) Korteweg-de Vries, NonLinear Schrödinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzéica, (Ward's) harmonic map eqs., and so on. This would guarantee existence of twistor description of them and the corresponding physical situations in N=2 string theory, and lead to fruitful applications to noncommutative integrable systems and string theories. Some integrable aspects of them are also discussed.

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“…Imposing the closure, one gets strong constraints on the integers m, r and s namely m = 0 r ≤ s ≤ 1 (27) With these constraint equations, the sub-spaces Σ (r,s) m exhibit then a Lie algebra structure since the ⋆-product is associative.…”

Section: Further Algebraic Properties Ofmentioning

confidence: 99%

“…As discussed previously, the subspaces Σ (r,s) m exhibit a Lie algebra structure with respect to the Moyal bracket once the spin-degrees constraints (27) are considered. With these conditions one should note that the huge Lie algebra that we can extract from the space Σ having the remarkable space decomposition …”

Section: The Lie Algebramentioning

confidence: 99%

“…Using the convention notations and the analysis presented previously and developed in [20,21] and based also on the results established in [25,26,27] 6 , we present in this section some results related to the Lax representation of noncommutative integrable hierarchies. We perform also consistent algebraic computations, based on the Moyal-momentum analysis, to derive explicit Lax pair operators of some integrable systems in the noncommutative framework.…”

Section: The Nc Lax Generating Technics In the Moyal Momentum Frameworkmentioning

confidence: 99%

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Moyal noncommutative integrability and the Burgers–KdV mapping

Sedra

2006

Nuclear Physics B

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The Moyal momentum algebra, studied in [20,21], is once again used to discuss some important aspects of NC integrable models and 2d conformal field theories. Among the results presented, we setup algebraic structures and makes useful convention notations leading to extract non trivial properties of the Moyal momentum algebra. We study also the Lax pair building mechanism for particular examples namely, the noncommutative KdV and Burgers systems. We show in a crucial step that these two systems are mapped to each others through the following crucialx . This makes a strong constraint on the NC Burgers system which corresponds to linearizing its associated differential equation. From the CFT's point of view, this constraint equation is nothing but the analogue of the conservation law of the conformal current. We believe that the considered mapping might help to bring new insights towards understanding the integrability of noncommutative 2d-systems.

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Extensions of Soliton equations to non-commutative (2 + 1) dimensions

Cited by 13 publications

References 16 publications

Noncommutative Solitons and Integrable Systems

Noncommutative Solitons and Integrable Systems

Non-commutative Ward's conjecture and integrable systems

Moyal noncommutative integrability and the Burgers–KdV mapping

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