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.

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:

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

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:

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), 325205.

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 Ha.

Nondegenerate solutions

One can show that

and

Construction of a

It satisfies

Construction of the free energy

Then

Special case

~ Thank You ~