Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

TL;DR

This paper constructs explicit series solutions for the dispersionless 2D Toda hierarchy, relates them to Hurwitz numbers, and explores applications in conformal dynamics and inverse problems.

Contribution

It introduces symmetric solutions to the dispersionless 2D Toda hierarchy, expresses coefficients via combinatorial constants, and derives new formulas for genus 0 double Hurwitz numbers.

Findings

Explicit series expansion for symmetric solutions

Recurrence relations for Taylor coefficients

New formulas for genus 0 double Hurwitz numbers

Abstract

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.