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Adler, Mark; Moerbeke, Pierre van

doi:10.1155/s1073792801000460

Cited by 34 publications

(13 citation statements)

References 14 publications

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“…See also the papers [19,20] and [72] on the multicomponent KP hierarchy and [81] on the multi-component Toda lattice hierarchy. In a series of papers Mark Adler and Pierre van Moerbeke showed how the Gauss-Borel factorization problem appears in the theory of the 2D Toda hierarchy and what they called the discrete KP hierarchy [1,2,3,4,5,6]. These papers clearly established -from a group-theoretical setup-why standard orthogonality of polynomials and integrability of nonlinear equations of Toda type where so close.…”

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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“…It also has been proven in [11] that operators of the form D = ∂ 2 F 2 (t) + ∂ 1 F 1 (t) + ∂ 0 F 0 have as eigenfunctions different infinite families of MOP's. In [12,13] matrix extensions of the generalized polynomials considered in [14,15] were studied. Recently, in [16], the Christoffel transformation to matrix orthogonal polynomials in the real line (MOPRL) has been extended and a new matrix Christoffel formula was obtained.…”

Section: Introductionmentioning

confidence: 99%

Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

Branquinho

Foulquié-Moreno²,

Fradi³

et al. 2022

Mathematics

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In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

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“…In [29], the two Darboux transforms on band matrices called LU and U L Darboux transformations are constructed, particularly for the 2m + 1-band matrix. In fact the 2m + 1-band matrix corresponds to the (m, m)-EBTH.…”

Section: Introductionmentioning

confidence: 99%

Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

Song

2016

Zeitschrift Für Naturforschung A

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Abstract. With the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy(EBTH) was proved to govern the Gromov-Witten theory of orbiford c N M in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this paper, the multi-fold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from known solutions which include soliton solutions of (N, N )-EBTH, i.e. the EBTH when N = M . From the generation of new solutions, one can find the big difference between the EBTH and the extended Toda hierarchy(ETH). Meanwhile we plotted the soliton graphs of the (N, N )-EBTH from which some approximation analysis will be given. From the analysis on velocities of soliton solutions, the difference between the extended flows and other flows are shown. The two different Darboux transformations constructed by us might be useful in Gromov-Witten theory of orbiford c N M . Mathematics Subject Classifications(2000). 37K10, 37K20.

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Cited by 34 publications

References 14 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

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