Article
The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and RiemannHilbert problems
Inverse Problems (Impact Factor: 1.32). 05/2010; 26(5). DOI: 10.1088/02665611/26/5/055009
Source: OAI
ABSTRACT
15 pages. ArXiv preprint available at: http://arxiv.org/abs/0911.0941 Submitted to: Inverse Problems We consider the relation of the multicomponent 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multigraded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a RiemannHilbert problem. MM thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS200800200 and UF thanks economical support from the Spanish Ministerio de Ciencia e Innovación research projects MTM200613000C0302 and MTM200762945 and from Comunidad de Madrid/Universidad Carlos III de Madrid project CCG07UC3M/ESP3339. No publicado
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 "One can also consider the bigraded extension of the extended QTH which might be included in our future work. The multicomponent 2D Toda hierarchy was considered from the point of view of the Gauss–Borel factorization problem, nonintersecting Brownian motions and matrix RiemannHilbert problem [31] [32] [33] [34]. In fact the http://dx.doi.org/10.1016/j.chaos.2015.03. "
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ABSTRACT: In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new qdeformed Toda hierarchy (QTH). Meanwhile the Block type additional symmetry and biHamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the qToda hierarchy to an extended qToda hierarchy (EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov–Witten theory. The corresponding Sato theory including multifold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the qToda hierarchy and show the integrability including its biHamiltonian structure, tau symmetry and conserved densities.  [Show abstract] [Hide abstract]
ABSTRACT: Multiple orthogonality is considered in the realm of a GaussBorel factorization problem for a semiinfinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of MNikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a GaussBorel factorization problem for a semiinfinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are biorthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multidiagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the ChristoffelDarboux formula are rederived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multicomponent 2D Toda hierarchy, which can be also understood and studied through a GaussBorel factorization problem, is discussed. Deformations of the weights, natural for MNikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and ZakharovShabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the $\tau$function representation of the multiple orthogonality is given.  [Show abstract] [Hide abstract]
ABSTRACT: We give a RiemannHilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of secondorder differential equations satisfied by the orthogonal polynomials, some of them of 0th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.