Analytic study of Painleve equations and a nelated clans of equation

Painleve方程及相关方程族的解析研究

• 批准号: 17340050
• 负责人: SHIMOMURA Shun
• 金额: $ 3.23万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340050/
• 关键词: Painleve equations; Ganier system; quasi-Painleve property; Fibonacci numbers; Painleve方程式; Garnier系; 準Painleve性; 特異点; Painleve equation; 準パンルヴェ性

1. We found a class of nonlinear ordinary differential equations containing the first Painleve equation such that each equation admits the quasi-Painleve property, and discussed the global multi-valuedness of their solutions. We also found a similar clean of equations containing the second Painleve equation.2. For solutions of the fifth Painleve equation, we examined value distribution in a sectonial domain. Under a certain condition, we proved the equi-distribution of values.3. For a degenerate Garnier system corresponding to the first Painleve equation, we obtain an asymptotic solution around a singular locus.4. We examined algebraic independence and algebraic relations for reciprocal rums of Fibonacci numbers. We also discussed their relation to zeta values.

1.找到了一类含有第一类Painleve方程的非线性常微分方程解的拟Painleve性质，并讨论了其解的整体多值性。我们还发现了一个类似的包含第二个Painleve方程的方程式。对于第五个Painleve方程的解，我们研究了在分段区域中的值分布。在一定条件下，证明了值的均匀分布。对于对应于第一个Painleve方程的退化的Garnier系统，我们得到了它在奇异轨迹附近的渐近解。我们研究了Fibonacci数的倒数的代数独立性和代数关系。我们还讨论了它们与Zeta值的关系。

项目成果

A Class of diffecestial equationd of PI-type toith the guasi-painleup propcity

一类具有偏-Painleup性质的PI型微分方程

• 发表时间: 2007
• 期刊: Aun.Mat.Pusa Appl 186
• 作者: K.Tanaka;P.Felmer;S.Martinez;Takaaki Nishida;Tadashi Kawanago;Takaaki Nishida;西田 孝明;Hiroshi Kokubu;Takaaki Nishida;Shun Shimomura
• 通讯作者: Shun Shimomura

Algehaic nelations for reciprocal sums of binary recurrences

二元递推倒数和的代数关系

• 发表时间: 2006
• 期刊: Seminar on Mathematical Sciences, Diophanting Ancly513 and Related Fiedds 2006 35
• 作者: Carsten Elsmer;Shun Shimomura;Iehata shiokawa
• 通讯作者: Iehata shiokawa

Asymptoric solutions of a degenerate Garnier system of the first Painleve type

第一 Painleve 型简并 Garnier 系统的渐近解

• 发表时间: 2007
• 期刊: RIMS Kokyuroku Bessatsu B2
• 作者: K.Tanaka;P.Felmer;S.Martinez;Takaaki Nishida;Tadashi Kawanago;Takaaki Nishida;西田 孝明;Hiroshi Kokubu;Takaaki Nishida;Shun Shimomura;Shun Shimomura
• 通讯作者: Shun Shimomura

A class of differential equation of PI-type with the quasi-Painle' property

一类具有拟Painle性质的PI型微分方程

• 期刊: Ann.Mat.Pura Appl. (to appear)
• 作者: Carsten Elsmer;Shun Shimomura;Iehata shiokawa;Shun Shimomura
• 通讯作者: Shun Shimomura