Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations

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 (∞) = ∞, f (0) = 0 and f′(0)g′(∞) = 1. For each solution, we show how to define the natural time variables ${t_n, n\in\mathbb{Z}}$, 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 Σ of ${\mathfrak{D}}$ defined by ${f(w)=1/\overline{g(1/\bar{w})}}$. This recovers the result of Zabrodin (Theor Math Phy 12:1511–1525, 2001).