Download presentation

Presentation is loading. Please wait.

Published byElise Keyt Modified over 2 years ago

1
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.

2
Dispersionless Toda Hierarchy Dispersionless 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:

3
where The Poisson bracket is defined by

4
The corresponding Orlov-Schulman functions are They satisfy the following evolution equations: Moreover, the following canonical relations hold:

5
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

6
The generalized Grunsky coefficients are defined by They can be compactly written as

7
Hence,

8
It follows that

9
Given a solution of the dispersionless Toda hierarchy, there exists a phi function and a tau function such that Identifying then Tau Functions

10
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

11
Nondegenerate Soltuions If and therefore Hence, then Such a solution is said to be degenerate.

12
If Then

13
Hence,

14
We find that and we have the generalized string equation: Such a solution is said to be nondegenerate.

16
Let Define

17
One can show that

18
Define Proposition:

19
where

20
is a function such that

21
Hence,

22
Let Then

23
We find that

24
Hence, Similarly,

25
Special Case

26
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.

27
Universal Whitham Hierarchy Lax equations:

28
Orlov-Schulman functions They satisfy the following Lax equations and the canonical relations

29
where They have Laurent expansions of the form

30
we have From

31
In particular,

32
Hence, and

33
The free energy F is defined by Free energy

34
Generalized Faber polynomials and Grunsky coefficients Notice that

35
The generalized Grunsky coefficients are defined by

36
The definition of the free energy implies that

37
Riemann-Hilbert Data: Nondegeneracy implies that for some function H a.

38
Nondegenerate solutions

39
One can show that and

40
Construction of a It satisfies

41
Construction of the free energy Then

42
Special case

Similar presentations

Presentation is loading. Please wait....

OK

Completeness and Expressiveness

Completeness and Expressiveness

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on image demosaicing Ppt on unity in diversity studio Ppt on natural resources and wildlife Ppt on principles of peace building organizations Ppt on personality development for college students Ppt on nursing management of spinal cord injury Ppt on human chromosomes chart Ppt on solar energy and its applications Ppt on unipolar nrz Ppt on soil pollution and its control