Dromion solutions of noncommutative Davey–Stewartson equations

Gilson, Claire R.; Macfarlane, Susan R.

doi:10.1088/1751-8113/42/23/235202

Cited by 36 publications

(25 citation statements)

References 21 publications

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“…Let us mention that in the early 1990 the Gelf'and school [51] already noticed the role quasi-determinants for some integrable systems, see also [94] for some recent work in this direction regarding non-Abelian Toda and Painlevé II equations. Jon Nimmo and his collaborators, the Glasgow school, have studied the relation of quasi-determinants and integrable systems, in particular we can mention the papers [55,56,67,54,68]; in this direction see also [58,124,59]. All this paved the route, using the connection with orthogonal polynomials a la Cholesky, to the appearance of quasi-determinants in the multivariate orthogonality context.…”

Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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Abstract. Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, its second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants -as well as Schur complements-of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R D , of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involves only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials are given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

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Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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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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“…(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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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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Dromion solutions of noncommutative Davey–Stewartson equations

Cited by 36 publications

References 21 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

Soliton Scattering in Noncommutative Spaces

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

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