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## Non-degenerate solutions of universal Whitham hierarchy

Journal of Physics A: Mathematical and Theoretical. 2010. Vol. 43. No. 325205. P. 1-22.

The notion of non-degenerate solutions for the dispersionless Toda hierarchy is
generalized to the universalWhitham hierarchy of genus zerowithM+1marked
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
(zα(p) : α = 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.

Research target:
Philosophy, Ethics, and Religious Studies

Language:
English