nondegenerate solutions of dispersionless toda hierarchy and tau functions teo lee peng university...
TRANSCRIPT
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 ~