Non-degenerate solutions of universal Whitham hierarchy

TL;DR

This paper extends the concept of non-degenerate solutions from the dispersionless Toda hierarchy to the universal Whitham hierarchy of genus zero, characterizing solutions via a Riemann-Hilbert problem involving canonical transformations and period maps.

Contribution

It introduces a generalized framework for non-degenerate solutions of the universal Whitham hierarchy using Riemann-Hilbert problems and conformal maps, expanding the understanding of hierarchy solutions.

Findings

Solutions characterized by Riemann-Hilbert problems with arbitrary generating functions.

Period maps defined by contour integrals generalize harmonic moments.

The free energy function has a contour integral representation.

Abstract

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 $H_a(z_0,z_a)$, $a = 1,...,M$, 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 $(z_\alpha(p) : \alpha = 0,1,...,M)$ 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.