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 $g(\infty )=\infty$, $f(0)=0$ and $f^{\prime }(0)g^{\prime }(\infty )=1$. 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 $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.

25 pages

References in corpus (6)

• 2D growth processes: SLE and Loewner chains
• Normal random matrix ensemble as a growth problem
• Growth processes related to the dispersionless Lax equations
• Genus-zero Whitham hierarchies in conformal-map dynamics
• Conformal Mappings and Dispersionless Toda hierarchy
• Multi-Cuts Solutions of Laplacian Growth

Cited by in corpus (4)

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• Canonical and grand canonical partition functions of Dyson gases as tau-functions of integrable hierarchies and their fermionic realization
• Universal Schlesinger system and Birkhoff factorization