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 ~