Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions

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

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