Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations

TL;DR

This paper classifies solutions to the dispersionless Toda hierarchy, linking them to conformal mappings and providing explicit formulas for the tau function, with implications for integrable systems and complex analysis.

Contribution

It introduces a classification of solutions based on a generating function and derives explicit tau function formulas, extending the understanding of the hierarchy's geometric structure.

Findings

Non-degenerate solutions determined by a function al H(z_1,z_2)

Explicit tau function formulas in terms of al H(z_1,z_2)

Restriction of flows to a subspace recovering known results

Abstract

In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We interpret these non-degenerate solutions as defining evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$, where $g$ is a univalent function on the exterior of the unit disc, $f$ is a univalent function on the unit disc, normalized such that $g(\infty)=\infty$, $f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the natural time variables $t_n, n\in\Z$, as complex coordinates on the space $\mathfrak{D}$. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.