Movable algebraic singularities of second-order ordinary differential equations
Movable algebraic singularities of second-order ordinary differential equations
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二阶常微分方程的动代数奇点
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
R. Halburd
中科院分区：
文献类型：
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作者：
G. Filipuk;R. Halburd
任何形式为y″=∑n= 0 Nan（z）yn的非线性方程都有一个解，其超前性态在点z 0附近与（z-z 0）−2/（N−1）成比例，其中系数an在z 0处解析，aN（z 0）<$0。我们考虑这样的方程，其中每一个可能的这种形式的首项都可以延伸到z− z 0的分数幂的罗朗级数解。对于这些方程，我们表明，只有可移动的奇点，可以通过解析延拓沿着有限长度的曲线是刚才所述的代数类型。这推广了Shimomura [“On second order nonlinear differential equations with the quasi-Painleve property II，”RIMS Kokyuroku 1424，177（2005）]的结果。这些代数奇点可以积累的可能性，沿着无限长的路径结束于一个有限的点被认为是。Smith [“On the singularities in the complex plane of the solutions of y″+y′f（y）+g（y）=P（x），”Proc. Lond. [Math.Soc.3，498（1953）]表明，这种奇异性确实发生在一个简单方程的解中......
Any nonlinear equation of the form y″=∑n=0Nan(z)yn has a solution with leading behavior proportional to (z−z0)−2/(N−1) about a point z0, where the coefficients an are analytic at z0 and aN(z0)≠0. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of z−z0. For these equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This generalizes results of Shimomura [“On second order nonlinear differential equations with the quasi-Painleve property II,” RIMS Kokyuroku 1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith [“On the singularities in the complex plane of the solutions of y″+y′f(y)+g(y)=P(x),” Proc. Lond. Math. Soc. 3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation out...