arXiv 0906.35652009-06-19math-phmath.MP

共形映射与无色散Toda族II：一般弦方程

Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations

Lee-Peng Teo

备注：: 25 pages
发表信息：: Commun.Math.Phys.297:447-474,2010
DOI：: 10.1007/s00220-010-1040-9

摘要：

在这篇文章中，我们将无色散Toda族的解分为退化和非退化两种情况。我们证明了每一个非退化的解决方案是由一个功能$\mathcal{H}(z_1,z_2)$的两个变量。我们解释这些非退化的解决方案定义的空间$\mathfrak{D}$上的共形映射$(g,f)$对，其中$g$是一个单叶函数的外部的单位盘，$f$是一个单叶函数的单位盘，规范化，使$g(\infty)=\infty$，$f(0)=0$和$f'(0)g'(\infty)=1$。对于每个解决方案，我们将展示如何定义自然时间变量$t_n, n\in\Z$，作为空间$\mathfrak{D}$上的复坐标。我们还发现明确的公式的τ函数的色散Toda系列的$\mathcal{H}(z_1, z_2)$。通过对函数$\mathcal{H}(z_1, z_2)$施加一定的条件，我们证明了无色散Toda流可以自然地限制在由$f(w)=1/\overline{g(1/\bar{w})}$定义的$\mathfrak{D}$的子空间$Σ$中.这恢复了Zabrodin的结果。

英文摘要：

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 $Σ$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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