The<i>U</i>(<i>N</i>) chiral model and exact multi-solitons

Haider, Bushra; Hassan, M.

doi:10.1088/1751-8113/41/25/255202

Cited by 19 publications

(12 citation statements)

References 47 publications

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“…Noncommutative extension of it would be possible and interesting (cf. [16]). Noncommutative binary Darboux transformations are successful for KP and KdV equations [13] and modified KP equations [15].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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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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Section: Conclusion and Discussionmentioning

confidence: 99%

“…(See, e.g. [12,13,14,15,16,19,29,41] and references therein.) For example, the noncommutative potential KP equation is derived in the framework of the noncommutative integrable hierarchy as follows:…”

Section: Introductionmentioning

confidence: 99%

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

Gilson¹,

Hamanaka²,

Nimmo³

2009

Glasgow Math. J.

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“…(See e.g. [10,12,15,16,17,18,19,20,21,27,28,37,48,50] and references therein.) It is interesting that quasideterminants simplify proofs in commutative theories.…”

Section: Introductionmentioning

confidence: 99%

Soliton Scattering in Noncommutative Spaces

Hamanaka¹,

Okabe²

2018

Theor Math Phys

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We discuss exact multi-soliton solutions to integrable hierarchies on noncommutative space-times in diverse dimension. The solutions are represented by quasi-determinants in compact forms. We study soliton scattering processes in the asymptotic region where the configurations could be real-valued. We find that the asymptotic configurations in the soliton scatterings can be all the same as commutative ones, that is, the configuration of N-soliton solution has N isolated localized lump of energy and each solitary wave-packet lump preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. As new results, we present multi-soliton solutions to noncommutative anti-self-dual Yang-Mills hierarchy and discuss 2-soliton scattering in detail.

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“…We first give a brief review of the elementary binary Darboux transformation of the chiral model (see [13][14][15][16][17][18][19][20][21][22][23][24] for details).…”

Section: Elementary Binary Darboux Transformationmentioning

confidence: 99%