Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2018.104 ◽

2018 ◽

Author(s):

Mattia Cafasso ◽

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Ann du Crest de Villeneuve ◽

Di Yang ◽

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...

Keyword(s):

Schur Polynomials ◽

Tau Functions

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On tau-functions for the KdV hierarchy

Selecta Mathematica ◽

10.1007/s00029-021-00620-x ◽

2021 ◽

Vol 27 (1) ◽

Author(s):

Boris Dubrovin ◽

Di Yang ◽

Don Zagier

Keyword(s):

Tau Functions ◽

Kdv Hierarchy

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Padé approximants on Riemann surfaces and KP tau functions

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00585-2 ◽

2021 ◽

Vol 11 (4) ◽

Author(s):

Marco Bertola

Keyword(s):

Riemann Surface ◽

Riemann Surfaces ◽

Hilbert Problem ◽

Line Bundles ◽

Fixed Degree ◽

Tau Function ◽

Tau Functions ◽

Deformation Parameters ◽

Higher Genus ◽

Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

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