Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis
Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis
复制标题
复分析若干问题中Hirota方程的无色散极限
DOI：
10.1023/a:1012883123413
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发表时间：
2001
影响因子：
1
通讯作者：
A. Zabrodin
中科院分区：
文献类型：
--
作者：
A. Zabrodin
本文研究了单复变函数理论中一些经典问题中最近发现的可积结构。给定复平面上以简单解析曲线为界的单连通区域，研究了与该区域相关的保角映射问题、Dirichlet边界问题和二维逆势问题。在这些域的空间上构造了一组显著的实值泛函。将该族中的任意泛函视为具有无穷多变量的函数，这些变量是定义域上适当定义的矩，给出上述问题的形式解。这些函数满足无穷多的无色散Hirota方程，因此是可积层次的tau函数。该层次结构与二维Toda链的无色散极限相一致。除了我们之前的研究之外，我们还表明，在力矩的更一般定义中，这种联系不属于Hirota方程的特解，而是属于层次本身。
We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.