Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation

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

doi:10.1098/rspa.2009.0647

Cited by 26 publications

(35 citation statements)

References 39 publications

(98 reference statements)

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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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“…In order to construct the multisoliton solution of the SUSY HM model, we define the Darboux transformation on the superfields that provides the generalization of Darboux transformation to the supersymmetric case. In our case, the Darboux transformation is defined by N×N superfield matrix  q x l ( ) x t , , , , called the superfield Darboux matrix (for detail discussion on the Darboux transformation we can see [14][15][16][17][18][19][20][21][22][23][24][25][26]). Now, let us define the superfied matrix  q x l ( ) x t , , , ; , such that superfield  of the Lax pair transforms as…”

Section: Darboux Transformation Of Supersymmetric Hm Model and Multismentioning

confidence: 99%

Darboux transformations of supersymmetric Heisenberg magnet model

Amjad

Haider

2018

J. Phys. Commun.

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“…New examples of the Darboux transformations for various cases have been appearing (see, e.g. [1][2][3]). …”

Section: Introductionmentioning

confidence: 98%

Invertible Darboux transformations of type I

Shemyakova

2015

Program Comput Soft

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In the paper, a general category based approach to invertible Darboux transformations is sug gested. For operators of arbitrary dimension and arbitrary order, it is suggested to consider Darboux transfor mations of certain type, namely, of type I. Explicit compact formulas are obtained. In particular, any invert ible Darboux transformations of first order and, for example, Laplace transformations are type I transforma tions. For operators of third order in the plane, analogues of the Darboux transformation chains are obtained.

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Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation

Cited by 26 publications

References 39 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

Darboux transformations of supersymmetric Heisenberg magnet model

Invertible Darboux transformations of type I

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