Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus LP Teo Conformal Mappings and Dispersionless Toda hierarchy II General String Equations ID: 629992
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Slide1
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.Slide2
Dispersionless Toda HierarchyDispersionless
Toda hierarchy describes the evolutions of two formal power series:
with respect to an infinite set of time variables
t
n
,
n
Z. The evolutions are determined by the Lax equations:Slide3
where
The Poisson bracket is defined bySlide4
The corresponding Orlov-Schulman functions are
They satisfy the following evolution equations:
Moreover, the following canonical relations hold:Slide5
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 Slide6
The generalized Grunsky coefficients are defined by
They can be compactly written asSlide7
Hence, Slide8
It follows thatSlide9
Given a solution of the dispersionless Toda hierarchy, there exists a phi function and a tau function such that
Identifying
then
Tau FunctionsSlide10
Riemann-Hilbert DataThe 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 relationSlide11
Nondegenerate SoltuionsIf
and therefore
Hence,
then
Such a solution is said to be
degenerate
. Slide12
If
ThenSlide13
Then
Hence,Slide14
We find that
a
nd we have the generalized string equation:
Such a solution is said to be
nondegenerate
. Slide15Slide16
Let
Define Slide17
One can show thatSlide18
Define
Proposition:Slide19
Proposition:
whereSlide20
i
s a function such thatSlide21
Hence,Slide22
Let
ThenSlide23
We find thatSlide24
Hence,
Similarly,Slide25
Special CaseSlide26
Generalization to Universal Whitham HierarchyK. Takasaki, T. Takebe and L. P. Teo, “Non-degenerate solutions of universal Whitham hierarchy”, J. Phys. A
43 (2010), 325205.Slide27
Universal Whitham Hierarchy
Lax equations:Slide28
Orlov-Schulman functions
They satisfy the following Lax equations
and the canonical relations Slide29
where
They have Laurent expansions of the formSlide30
we have
FromSlide31
In particular, Slide32
Hence,andSlide33
The free energy F is defined by
Free energySlide34
Generalized Faber polynomials and Grunsky coefficients
Notice thatSlide35
The generalized Grunsky coefficients are defined bySlide36
The definition of the free energy implies thatSlide37
Riemann-Hilbert Data:
Nondegeneracy
implies that
for some function
H
a
.Slide38
Nondegenerate solutionsSlide39
One can show that
andSlide40
Construction of a
It satisfiesSlide41
Construction of the free energy
ThenSlide42
Special caseSlide43
~ Thank You ~