Non-degenerate solutions of the universal Whitham hierarchy

The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with M + 1 marked points. These solutions are characterized by a Riemann--Hilbert problem (generalized string equations) with respect to two-dimensional canoni...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2010-08, Vol.43 (32), p.325205-325205
Main Authors: Takasaki, Kanehisa, Takebe, Takashi, Teo, Lee Peng
Format: Article
Language:English
Subjects:
Free energy
Hierarchies
Integrals
Mathematical analysis
Shape
Transformations
Two dimensional
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Summary:The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with M + 1 marked points. These solutions are characterized by a Riemann--Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations and may be thought of as a kind of general solutions of the hierarchy. The Riemann--Hilbert problem contains M arbitrary functions Ha(z0, za), a = 1, ..., M, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riemann--Hilbert problem is described by period maps on the space of (M + 1)-tuples (z Delta *a(p): Delta *a = 0, 1, ..., M) of conformal maps from M disks of the Riemann sphere and their complements to the Riemann sphere. The period maps are defined by an infinite number of contour integrals that generalize the notion of harmonic moments. The F-function (free energy) of these solutions is also shown to have a contour integral representation.
ISSN:1751-8121
1751-8113
1751-8121
DOI:10.1088/1751-8113/43/32/325205