nondegenerate solutions of dispersionless toda hierarchy and tau functions teo lee peng university...
TRANSCRIPT
![Page 1: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/1.jpg)
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:
![Page 3: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/3.jpg)
where
The Poisson bracket is defined by
![Page 4: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/4.jpg)
The corresponding Orlov-Schulman functions are
They satisfy the following evolution equations:
Moreover, the following canonical relations hold:
![Page 5: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/5.jpg)
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
![Page 6: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/6.jpg)
The generalized Grunsky coefficients are defined by
They can be compactly written as
![Page 7: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/7.jpg)
Hence,
![Page 8: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/8.jpg)
It follows that
![Page 9: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/9.jpg)
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
![Page 11: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/11.jpg)
Nondegenerate Soltuions
If
and therefore
Hence,
then
Such a solution is said to be degenerate.
![Page 12: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/12.jpg)
If
Then
![Page 13: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/13.jpg)
Then
Hence,
![Page 14: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/14.jpg)
We find that
and we have the generalized string equation:
Such a solution is said to be nondegenerate.
![Page 15: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/15.jpg)
![Page 16: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/16.jpg)
Let
Define
![Page 17: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/17.jpg)
One can show that
![Page 18: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/18.jpg)
Define
Proposition:
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Proposition:
where
![Page 20: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/20.jpg)
is a function such that
![Page 21: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/21.jpg)
Hence,
![Page 22: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/22.jpg)
Let
Then
![Page 23: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/23.jpg)
We find that
![Page 24: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/24.jpg)
Hence,
Similarly,
![Page 25: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/25.jpg)
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
![Page 29: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/29.jpg)
where
They have Laurent expansions of the form
![Page 30: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/30.jpg)
we have
From
![Page 31: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/31.jpg)
In particular,
![Page 32: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/32.jpg)
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
![Page 36: Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings](https://reader030.vdocuments.us/reader030/viewer/2022032517/56649cae5503460f94970e0f/html5/thumbnails/36.jpg)
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 ~