Non-local continuity equations and binary Darboux transformation of noncommutative (anti) self-dual Yang–Mills equations

Saleem, U.; Hassan, M.; Siddiq, M.

doi:10.1088/1751-8113/40/19/019

Cited by 6 publications

(13 citation statements)

References 63 publications

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…This gauge is suitable for the discussion of (binary) Darboux transformations for (noncommutative) ASDYM equation (Gilson, Nimmo and Ohta (1998); Nimmo, Gilson and Ohta (2000); Saleem, Hassan and Siddiq (2007)).…”

Section: Gauge Transformations Act On H and H Asmentioning

confidence: 99%

“…Furthermore investigation of the noncommutative extension of a bilinear form approach to the ASDYM equation (Gilson, Nimmo and Ohta (1998); Sasa, Ohta and Matsukidaira (1998); Wang and Wadati (2004)) would be beneficial because many aspects in these paper are close to ours. The relationship with noncommutative Darboux and noncommutative binary Darboux transformations (Salaam, Hassan and Siddiq (2007)) is also interesting. quasideterminant associated with the bottom right element is simply S. By choosing an appropriate partitioning, any entry in the inverse of a square matrix can be expressed as the inverse of a Schur complement and hence quasideterminants can also be defined recursively by:…”

Section: Twistor Descriptions Of the Noncommutative Asdym Equationsmentioning

confidence: 99%

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Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

2009

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We present Bäcklund transformations for the non-commutative anti-self-dual YangMills equation where the gauge group is G = GL(2), and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants. We also explain the origins of all the ingredients of the Bäcklund transformations within the framework of non-commutative twistor theory. In particular, we show that the generated solutions belong to a non-commutative version of the Atiyah-Ward ansatz.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Gauge Transformations Act On H and H Asmentioning

confidence: 99%

Section: Twistor Descriptions Of the Noncommutative Asdym Equationsmentioning

confidence: 99%

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Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

2009

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“…Following [23], it is easy to see that nc-(A)SDYM equations (5.1) reduce to a noncommutative twodimensional principal chiral field equation, if we take y = ȳ = x 0 and z = z = x 1 ,…”

Section: Binary Darboux Transformation For a Noncommutative U(n) Prin...mentioning

confidence: 99%

“…The (A)SDYM equations also act as master equations of many integrable equations in the sense of Ward conjecture[48]-[50]. The noncommutative generalization of (A)SDYM equations and its integrability aspects have been investigated recently (e.g [21]-[23]…”

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confidence: 99%

TheU(N) chiral model and exact multi-solitons

Haider¹,

Hassan²

2008

J. Phys. A: Math. Theor.

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Bäcklund Transformations for Non-Commutative Anti-Self-Dual Yang–mills Equations

Gilson¹,

Hamanaka²,

Nimmo³

2009

Glasgow Math. J.

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Abstract. We present Bäcklund transformations for the non-commutative antiself-dual Yang-Mills equations where the gauge group is G = GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasi-determinants and belong to a noncommutative version of the Atiyah-Ward ansatz. In the commutative limit, our results coincide with those by Corrigan, Fairlie, Yates and Goddard.2000 Mathematics Subject Classification. 35Q58, 46L55, 70S15. Introduction.Non-commutative (NC) extensions of integrable systems and soliton theory have been studied intensively for the last few years in various contexts of both mathematics and physics (for reviews, see, e.g. [22]). In particular, the extension to NC spaces has drawn much attention in physics, because in gauge theories this kind of NC extension corresponds to the presence of background magnetic fields, and various applications have been made successfully (for reviews, see, e.g. [21]).In commutative gauge theories, the anti-self-dual Yang-Mills (ASDYM) equation is quite important. The finite-action solutions, that is the instanton solutions, play key roles in field theories and in four-dimensional geometry. In integrable systems, it is also very important since many lower-dimensional integrable equations such as the Korteweg-de Vries (KdV) [15], and therefore it is now worth studying the integrable aspects of the NC ASDYM equation in detail and applying the results to lower-dimensional integrable equations.

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Non-local continuity equations and binary Darboux transformation of noncommutative (anti) self-dual Yang–Mills equations

Cited by 6 publications

References 63 publications

Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

Bäcklund transformations and the Atiyah–Ward ansatz for non-commutative anti-self-dual Yang–Mills equations

TheU(N) chiral model and exact multi-solitons

Bäcklund Transformations for Non-Commutative Anti-Self-Dual Yang–mills Equations

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