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.
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.
evolutions are determined by the Lax equations:
The
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
They can be compactly written as
The generalized Grunsky coefficients are defined by
Hence,
It follows that
Tau Functions
Given a solution of the dispersionless Toda hierarchy, there exists
a phi function and a tau function such that
Identifying
then
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
then
Hence,
and therefore
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
From
we have
In particular,
Hence,
and
Free energy
The free energy F is defined by
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 ~