Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics

TL;DR

This paper explores symmetric solutions to the dispersionless Toda hierarchy, linking them to conformal mappings and fluid dynamics, and constructs a tau-function for inverse potential and Riemann mapping problems.

Contribution

It introduces a class of symmetric solutions depending on radial density functions and constructs a tau-function connecting to conformal and fluid flow problems.

Findings

Constructed dispersionless tau-function for symmetric solutions

Linked solutions to inverse potential and Riemann mapping problems

Discussed conformal dynamics related to Hele-Shaw flows

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.