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),

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:

where The Poisson bracket is defined by

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

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

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

Hence,

It follows that

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

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

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

If Then

Hence,

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

Let Define

One can show that

Define Proposition:

where

is a function such that

Hence,

Let Then

We find that

Hence, Similarly,

Special Case

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),

Universal Whitham Hierarchy Lax equations:

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

where They have Laurent expansions of the form

we have From

In particular,

Hence, and

The free energy F is defined by Free energy

Generalized Faber polynomials and Grunsky coefficients Notice that

The generalized Grunsky coefficients are defined by

The definition of the free energy implies that

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

Nondegenerate solutions

One can show that and

Construction of  a It satisfies

Construction of the free energy Then

Special case