Nondegenerate Solutions of - PowerPoint Presentation

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

. Slide15
Slide16

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 ~