Noncommutative integrable systems and quasideterminants

Hamanaka, Masashi; Xiu, Wen; Hu, Xing−Biao; Liu, Qingping

doi:10.1063/1.3367025

Cited by 9 publications

(7 citation statements)

References 39 publications

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“…The noncommutative theory gives rise to various new physical objects in quantum mechanics such as the canonical commutation relation [q, p] = i . As said in [9], the noncommutative parameter is closely related to the existence of a background flux in the effective theory of D-branes. With the presence of background magnetic fields the noncommutative gauge theories were found to be equivalent to ordinary gauge theories and noncommutative solitons play important roles in the study of D-brane dynamics [10].…”

Section: Introductionmentioning

confidence: 89%

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

2018

Journal of Mathematical Physics

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In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili(KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra W 1+∞ . In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which contains many interesting noncommutative integrable systems such as the noncommutative KdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we construct two new noncommutative systems including odd noncommutative C type Gelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Also using the symmetry, we can construct a new noncommutative Gelfand-Dickey hierarchy with self-consistent sources. Basing on the natural differential Lax operator of the noncommutative Gelfand-Dickey hierarchy, the string equations of the noncommutative Gelfand-Dickey hierarchy are also derived. (2010): 37K05, 37K10, 35Q53. Mathematics Subject Classifications

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Section: Introductionmentioning

confidence: 89%

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

2018

Journal of Mathematical Physics

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“…The ordering of variables in non-linear terms is crucial to preserve some special integrable properties and determined in the Lax formalism. (For a review, see [3].) We note that the fields themselves take c-number values as usual and the differentiation and the integration for them are well-defined as usual, for example,…”

Section: Noncommutative Gauge Theoriesmentioning

confidence: 99%

Noncommutative solitons and quasideterminants

Hamanaka¹

2014

Phys. Scr.

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We discuss an extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang-Mills equations. We give a wide class of exact solutions by solving a Riemann-Hilbert problem for the Atiyah-Ward ansatz and present Bäcklund transformations for the G = U(2) noncommutative anti-self-dual Yang-Mills equations. We find that one kind of noncommutative determinant, quasideterminants, play crucial roles in the construction of noncommutative solutions. We also discuss reduction of a noncommutative anti-self-dual Yang-Mills equation to noncommutative integrable equations. This is partially based on a collaboration with C.

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“…Noncommutative spaces contain the spatial coordinates x i with the noncommutativity of [x i , x j ] = iθ i j , where the noncommutative parameter θ i j is a anti-symmetric tensor. As in [1], the noncommutative parameter is a real constant and closely related to the existence of a background flux. The noncommutative theory gives rise to various new physical objects such as the canonical commutation relation [q, p] = ih in quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

“…In the framework of noncommutative integrable hierarchy, infinite conserved quantities and exact soliton solutions were given for many noncommutative integrable equations in terms of Strachan's products and quasi-determinants. These noncommutative integrable equations have a close relation to an noncommutative anti-self-dual Yang-Mills theory [1]. An infinite number of conserved laws for the noncommutative Lax hierarchies were presented which include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equation and so on in [3].…”

Section: Introductionmentioning

confidence: 99%

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

Li¹,

Song²

2021

JNMP

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The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the noncommutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z N -Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.

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Noncommutative integrable systems and quasideterminants

Cited by 9 publications

References 39 publications

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

Noncommutative solitons and quasideterminants

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

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