由Painlevé级数显式构造含拟单项项的首次积分
Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlevé Series
摘要：
Painlevé图和弱Painlevé图已被广泛用于识别新的可积非线性动力系统。对于通过Painlevé检验的系统，积分的计算依赖于各种独立于Painlevé分析的方法。本文提出了一个明确的算法来建立一个动力系统的第一积分，表示为“准多项式”的功能，从所提供的信息完全由Painlevé - Laurent系列解决方案的常微分方程系统。应用Yoshida（1983）的一个定理，得到了对拟多项式积分中拟单项项的个数和形式的限制。通过适当平衡被选作积分的初始参数的准多项式函数中的系数来获得积分，从而消除了对时间$τ=t-t_0$的幂的所有依赖。左、右Painlevé级数在该方法中都是有用的。或者，该方法可以用来证明不存在拟多项式第一积分。从具体的动力系统的例子。
英文摘要：
The Painlevé and weak Painlevé conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlevé test, the calculation of the integrals relies on a variety of methods which are independent from Painlevé analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as `quasi-polynomial' functions, from the information provided solely by the Painlevé - Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial integral are obtained by an application of a theorem by Yoshida (1983). The integrals are obtained by a proper balancing of the coefficients in a quasi-polynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time $τ=t-t_0$ is eliminated. Both right and left Painlevé series are useful in the method. Alternatively, the method can be used to show the non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given.
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