On integrability of a noncommutative q -difference two-dimensional Toda lattice equation

Li, C.X.; Nimmo, J. J. C.; Shen, Shoufeng

doi:10.1016/j.physleta.2015.10.027

Cited by 7 publications

(9 citation statements)

References 34 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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“…We constructed its bilinear Bäcklund transformations and Lax pair by using Hirota's bilinear method. Besides, Darboux transformation was established to construct its quasi-Casoratian solutions for a noncommutative q-2DTL equation [1]. Actually, we can prove that the same Darboux transformation holds true for the commutative q-2DTL equation which can be used to reconstruct the existing Casoratian solutions to the bilinear q-2DTL equation and its bilinear Bäcklund transformation.…”

Section: Introductionmentioning

confidence: 80%

“…The nonlinear q-2DTL equation was first proposed in [21]. Later on, a slightly different nonlinear q-2DTL equation was presented in [1]. These two nonlinear equations correspond to the same bilinear equation under different dependent variable transformations.…”

Section: A Generalized Lax Pair For the Q-2dtl Equationmentioning

confidence: 99%

“…Denote a(x) = (q − 1)x and b(y) = (q − 1)y. In [1], a special bilinear Bäcklund transformation was presented for the q-2DTL equation (2.4). It was given by…”

Section: A Generalized Lax Pair For the Q-2dtl Equationmentioning

confidence: 99%

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confidence: 99%

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An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

Li¹,

Wang²,

Yao³

et al. 2022

Preprint

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In our previous work [1], we constructed quasi-Casoratian solutions to the noncommutative q-difference two-dimensional Toda lattice (q-2DTL) equation by Darboux transformation, which we can prove produces the existing Casoratian solutions to the bilinear q-2DTL equation obtained by Hirota's bilinear method in commutative setting. It is actually true that one can not only construct solutions to soliton equations but also solutions to their corresponding Bäcklund transformations by their Darboux transformations and binary Darboux transformations. To be more specific, eigenfunctions produced by iterating Darboux transformations and binary Darboux transformations for soliton equations give nothing but determinant solutions to their Bäcklund transformations, individually. This reveals the profound connections between Darboux transformations and Hirota's bilinear method. In this paper, we shall expound this viewpoint in the case of the q-2DTL equation. First, we derive a generalized bilinear Bäcklund transformation and thus a generalized Lax pair for the bilinear q-2DTL equation. And then we successfully construct the binary Darboux transformation for the q-2DTL equation, based on which, Grammian solutions expressed in terms of quantum integrals are established for both the bilinear q-2DTL equation and its bilinear Bäcklund transformation. In the end, by imposing the 2-periodic reductions on the corresponding results of the q-2DTL equation, we derive a qdifference sine-Gordon equation, a modified q-difference sine-Gordon equation and obtain their corresponding solutions.

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Darboux transformation for a semi-discrete matrix coupled dispersionless system

Riaz,

Lin

2024

Applied Mathematics Letters

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On integrability of a noncommutative q -difference two-dimensional Toda lattice equation

Cited by 7 publications

References 34 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

Darboux transformation for a semi-discrete matrix coupled dispersionless system

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