Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations. Articles uri icon

publication date

  • May 2018

start page

  • 1

end page

  • 46

issue

  • 20

volume

  • 51

international standard serial number (ISSN)

  • 1751-8113

electronic international standard serial number (EISSN)

  • 1751-8121

abstract

  • In this paper, Geronimus-Uvarov perturbations for matrix orthogonal polynomials on the real line are studied and then applied to the analysis of non-Abelian integrable hierarchies. The orthogonality is understood in full generality, i.e. in terms of a nondegenerate continuous sesquilinear form, determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. We derive Christoffel-type formulas that give the perturbed matrix biorthogonal polynomials and their norms in terms of the original ones. The keystone for this finding is the Gauss-Borel factorization of the Gram matrix. Geronimus-Uvarov transformations are considered in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies. The interplay between transformations and integrable flows is discussed. Miwa shifts, tau-ratio matrix functions and Sato formulas are given. Bilinear identities, involving Geronimus-Uvarov transformations, first for the Baker functions, then secondly for the biorthogonal polynomials and its second kind functions, and finally for the tau-ratio matrix functions, are found.

keywords

  • 2D non-Abelian Toda lattice hierarchy; noncommutative KP hierarchy; matrix biorthogonal polynomials; Christoffel-type formulas; quasideterminants; bilinear identities.