nondegenerate solutions of dispersionless toda hierarchy and tau functions
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Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions. Teo Lee Peng University of Nottingham Malaysia Campus. L.P. Teo, “Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations”, Commun . Math. Phys. 297 (2010), 447-474. - PowerPoint PPT PresentationTRANSCRIPT
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Nondegenerate Solutions of Dispersionless Toda Hierarchy
and Tau Functions
Teo Lee PengUniversity of Nottingham
Malaysia Campus
L.P. Teo, “Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations”, Commun. Math. Phys. 297 (2010), 447-474.
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Dispersionless Toda Hierarchy
Dispersionless Toda hierarchy describes the evolutions of two formal power series:
with respect to an infinite set of time variables tn, n Z. The evolutions are determined by the Lax equations:
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where
The Poisson bracket is defined by
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The corresponding Orlov-Schulman functions are
They satisfy the following evolution equations:
Moreover, the following canonical relations hold:
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Generalized Faber polynomials and Grunsky coefficients
Given a function univalent in a neighbourhood of the origin:
and a function univalent at infinity:
The generalized Faber polynomials are defined by
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The generalized Grunsky coefficients are defined by
They can be compactly written as
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Hence,
![Page 8: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions](https://reader033.vdocuments.us/reader033/viewer/2022051219/568167e4550346895ddd4c88/html5/thumbnails/8.jpg)
It follows that
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Given a solution of the dispersionless Toda hierarchy, there exists a phi function and a tau function such that
Identifying
then
Tau Functions
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Riemann-Hilbert Data
The Riemann-Hilbert data of a solution of the dispersionless Toda hierarchy is a pair of functions U and V such that
and the canonical Poisson relation
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Nondegenerate Soltuions
If
and therefore
Hence,
then
Such a solution is said to be degenerate.
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If
Then
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Then
Hence,
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We find that
and we have the generalized string equation:
Such a solution is said to be nondegenerate.
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Let
Define
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One can show that
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Define
Proposition:
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Proposition:
where
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is a function such that
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Hence,
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Let
Then
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We find that
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Hence,
Similarly,
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Special Case
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Generalization to Universal Whitham Hierarchy
K. Takasaki, T. Takebe and L. P. Teo, “Non-degenerate solutions of universal Whitham hierarchy”, J. Phys. A 43 (2010), 325205.
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Universal Whitham Hierarchy
Lax equations:
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Orlov-Schulman functions
They satisfy the following Lax equations
and the canonical relations
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where
They have Laurent expansions of the form
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we have
From
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In particular,
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Hence,
and
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The free energy F is defined by
Free energy
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Generalized Faber polynomials and Grunsky coefficients
Notice that
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The generalized Grunsky coefficients are defined by
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The definition of the free energy implies that
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Riemann-Hilbert Data:
Nondegeneracy
implies that
for some function Ha.
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Nondegenerate solutions
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One can show that
and
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Construction of a
It satisfies
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Construction of the free energy
Then
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Special case
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~ Thank You ~