论文标题
重量，Kovalevskaya指数和Painlevé物业
Weights, Kovalevskaya exponents and the Painlevé property
论文作者
论文摘要
研究了Parelevé方程的准杂种函数的加权程度。一个正整数的元组，称为正常的重量，满足与奇异理论有关的某些条件的分类。每个多项式Painlevé方程都有正常的重量。相反，对于$ 2 $和$ 4 $ -DIM的案例，表明存在一个差分方程，满足与每个常规重量相关的Painlevé属性。还通过常规权重，奇异理论和动力学系统理论研究了准汉密尔顿系统的Kovalevskaya指数。结果表明，通过系统爆破获得的laurent系列解决方案和相关向量场的稳定歧管之间存在一对一的对应关系。对于$ 4 $ -DIM的自主painlevé方程式，可以将哈密顿功能的水平表面分解为稳定的歧管的不相交结合。
Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlevé equations are investigated. A tuple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. Each polynomial Painlevé equation has a regular weight. Conversely, for $2$ and $4$-dim cases, it is shown that there exists a differential equation satisfying the Painlevé property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical systems theory. It is shown that there is a one-to-one correspondence between Laurent series solutions and stable manifolds of the associated vector field obtained by the blow-up of the system. For $4$-dim autonomous Painlevé equations, the level surface of Hamiltonian functions can be decomposed into a disjoint union of stable manifolds.
