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Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
Under certain reality conditions, a general solution to the dispersionless
Toda lattice hierarchy describes deformations of simply-connected plane domains
with a smooth boundary. The solution depends on an arbitrary (real positive)
function of two variables which plays the role of a density or a conformal
metric in the plane. We consider in detail the important class of symmetric
solutions characterized by the density functions that depend only on the
distance from the origin and that are positive and regular in an annulus . We construct the dispersionless tau-function which gives formal local
solution to the inverse potential problem and to the Riemann mapping problem
and discuss the associated conformal dynamics related to viscous flows in the
Hele-Shaw cell.Comment: 28 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1302.728
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