论文标题
Loewner方程和降低无分散层次结构
Loewner equations and reductions of dispersionless hierarchies
论文作者
论文摘要
Loewner类型的方程式可以在两个截然不同的上下文中得出：其中一个是复杂的分析和参数形象的理论，另一个是可集成系统的理论。在本文中，我们比较两种方法。在基于复杂分析的Löwner方程召回Löwner方程之后，我们回顾了无分散无整合的Hierarhies（DKP，DBKP，DTODA和DDKP）的单变量减少。一项可变的减少是通过不同版本的Loewner方程的溶液来描述的：DKP的弦（合理），DBKP的象限，radial（三角学）用于DTODA和DKP的椭圆形。我们还讨论了多变量的减少，这些减少是由由流体动力类型的部分微分方程补充的Loewner方程系统给出的。流体动力类型系统的溶解度可以通过广义的hodograph方法证明。
The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Löwner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of different versions of Loewner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP. We also discuss multi-variable reductions which are given by a system of Loewner equations supplemented by a system of partial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.
