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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.

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Dispersionless Toda Hierarchy Dispersionless Toda hierarchy describes the evolutions of two formal power series: with respect to an infinite set of time variables t n, n Z. The evolutions are determined by the Lax equations:

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where The Poisson bracket is defined by

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The corresponding Orlov-Schulman functions are They satisfy the following evolution equations: Moreover, the following canonical relations hold:

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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

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The generalized Grunsky coefficients are defined by They can be compactly written as

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Hence,

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It follows that

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Given a solution of the dispersionless Toda hierarchy, there exists a phi function and a tau function such that Identifying then Tau Functions

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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

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Nondegenerate Soltuions If and therefore Hence, then Such a solution is said to be degenerate.

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If Then

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Hence,

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We find that and we have the generalized string equation: Such a solution is said to be nondegenerate.

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Let Define

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One can show that

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Define Proposition:

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where

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is a function such that

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Hence,

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Let Then

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We find that

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Hence, Similarly,

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Special Case

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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.

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Universal Whitham Hierarchy Lax equations:

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Orlov-Schulman functions They satisfy the following Lax equations and the canonical relations

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where They have Laurent expansions of the form

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we have From

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In particular,

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Hence, and

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The free energy F is defined by Free energy

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Generalized Faber polynomials and Grunsky coefficients Notice that

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The generalized Grunsky coefficients are defined by

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The definition of the free energy implies that

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Riemann-Hilbert Data: Nondegeneracy implies that for some function H a.

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Nondegenerate solutions

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One can show that and

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Construction of a It satisfies

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Construction of the free energy Then

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Special case

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