Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics

Abstract

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 $r_0<|z|<r_1$. 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.

28 pages, 3 figures. arXiv admin note: text overlap with arXiv:1302.7288

References in corpus (6)

• Matrix Models for Random Partitions
• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Matrix models and growth processes: from viscous flows to the quantum Hall effect
• Growth processes related to the dispersionless Lax equations
• Laplacian growth in a channel and Hurwitz numbers
• Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics