Exact WKB analysis of higher order differential equations that is centered around the notion of a virtual turning point

以虚拟转折点概念为中心的高阶微分方程的精确 WKB 分析

基本信息

批准号：
17340035
负责人：
KAWAI Takahiro
金额：
$ 6.94万
依托单位：
Kyoto 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-17340035/
关键词：
Toulouse Project a higher order Painleve equation a virtual turning point an instanton-type solution exact WKB analysis a 0-parameter solution a turning point of the first kind a degenerate Garnier system ∞-Weber方程式形式変換ボレル平面動かない特異

项目摘要

Along the flow line described in "Toulouse Project', which is designed to clarify the structure of higher order Painleve equations, we have obtained the following results.1. A O-parameter solution of (P_J)m (J= I, II, IV) is locally and formally transformed to a 0-parameter solution of (P_I)_I near a turning point of the first kind.(T. Kawai and Y. Takei: Advances in Math., 203 (2006))2. Construction of a (2m)-parameter solution of instanton-type of (P_J)m (J= I, II, IV) (Y. Takei (2008), T. Koike: Kokyuroku Bessatsu, B2 (2007), B5 (2008))3. For each turning point of (P_J)m (J= I, U, IV) of the first kind, an instanton-type solution associated with the turning point can be locally and formally transformed to a 2-parameter solution of (P_I)_I. (T. Kawai and Y. Takei: Kokyuroku Bessatsu, B5 (2008))4. Each (P_J)m (J= I, II, IV) is isomorphic to an appropriate degenerate Gamier system restricted to some complex line. (T. Koike: Kokyuroku Bessatsu, B2 (2007), B5 (2008))In addition to the progress in the study of (P_J)m, we have made a substantial progress in the study of virtual turning points through the investigation of the Stokes geometry of another higher order Painleve equation called the Noumi-Yamada system. (P. Kawai et al. (2008)) Furthermore a novel treatment of several infinite series in exact WKB analysis with the aid of infinite order differential operator has been found by T. Aoki, T. Kawai and Y. Takei (RIMS Preprint 1616 (2007)).A conference centered around these results was held from January 28 through February 1, 2008 at CIRM (Centre International de Rencontres Mathematiques) in Marseilles (France). It turned out to be an intensive and fruitful meeting with many foreign (neither French nor Japanese) participants.

沿着“图卢兹计划”中描述的旨在阐明高阶 Painleve 方程结构的流程，我们得到了以下结果： 1. (P_J)m (J= I, II, IV) 在第一类转折点附近被局部形式化地转换为 (P_I)_I 的 0 参数解。（T. Kawai 和 Y. Takei：数学进展，203 (2006))2. (P_J)m 瞬子型 (J= I, II, IV) 的 (2m) 参数解的构造 (Y. Takei (2008), T. Koike: Kokyuroku Bessatsu, B2 ( 2007), B5 (2008))3. 对于第一类 (P_J)m (J= I, U, IV) 的每个转折点，有瞬子型解与转折点相关的可以局部且正式地转换为 (P_I)_I 的 2 参数解（T. Kawai 和 Y. Takei：Kokyuroku Bessatsu，B5 (2008)）4。 I、II、IV) 与受限于某些复杂系的适当简并 Gamier 系统同构（T. Koike：Kokyuroku Bessatsu，B2）。 (2007), B5 (2008)) 除了 (P_J)m 的研究进展外，我们通过研究另一个高阶 Painleve 方程（称为努海-山田系统。（P. Kawai 等人 (2008)）此外，T. Aoki、T. Kawai 和 Y. Takei 还发现了一种在精确 WKB 分析中借助无限阶微分算子对多个无穷级数进行新颖处理的方法（RIMS 预印本 1616） (2007))。围绕这些结果的会议于 2008 年 1 月 28 日至 2 月 1 日在 CIRM（国际数学竞赛中心）举行马赛（法国）。事实证明，这是一次密集且富有成效的会议，有许多外国（既不是法国也不是日本）参与者。