论文标题
潘leve系统的理性解决方案
Rational solutions of Painleve systems
论文作者
论文摘要
尽管painlevé方程的解决方案是不能以已知基本功能来表达的，但对于方程参数的专业值，确实存在理性解决方案。在研究Painlevé方程的理性解决方案的研究中，一种非常成功的方法涉及将这些标量方程式重新制定为一种耦合的，类似Riccati的方程式的对称系统，称为敷料链。已知定期调料链相当于Noumi和Yamada首先描述的$ A_N $-Painlevé系统。反过来，noumi-yamada系统可以线性化为使用双线性方程和$τ$ functions；然后可以将相应的理性解决方案作为KP层次结构的理性解决方案的专业提出。 现在可以将理性解决方案和系统的理性解决方案分类减少为对称为Maya图的组合对象的分析。该分析的结果是从经典正交多项式方面的理性解决方案的明确决定性表示。在本文中，我们通过描述Noumi-Yamada系统的painlevé的Hermite型理性解决方案来说明这种方法。举例来说，我们明确为PIV，PV方程和$ A_4 $PANELEVé系统构建了Hermite型解决方案。
Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very successful approach in the study of rational solutions to Painlevé equations involves the reformulation of these scalar equations into a symmetric system of coupled, Riccati-like equations known as dressing chains. Periodic dressing chains are known to be equivalent to the $A_N$-Painlevé system, first described by Noumi and Yamada. The Noumi-Yamada system, in turn, can be linearized as using bilinear equations and $τ$-functions; the corresponding rational solutions can then be given as specializations of rational solutions of the KP hierarchy. The classification of rational solutions to Painlevé equations and systems may now be reduced to an analysis of combinatorial objects known as Maya diagrams. The upshot of this analysis is a an explicit determinental representation for rational solutions in terms of classical orthogonal polynomials. In this paper we illustrate this approach by describing Hermite-type rational solutions of Painlevé of the Noumi-Yamada system in terms of cyclic Maya diagrams. By way of example we explicitly construct Hermite-type solutions for the PIV, PV equations and the $A_4$ Painlevé system.
