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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“Toulouse Project”中描述的旨在阐明高阶painlevel方程结构的流动线，我们得到了以下结果：（P_J）m （J= I， II， IV）的0参数解在第一类拐点附近局部形式转化为（P_I）_I的0参数解。Kawai和Y. Takei：《数学的进步》。， 2003 (2006)；2 . (P_J)m （J= I， II， IV）的瞬时型（2m）参数解的构造[Y. Takei (2008)， T. Koike: Kokyuroku Bessatsu, B2 (2007), B5(2008)]。对于第一类（P_J）m （J= I， U， IV）的每个拐点，与该拐点相关的瞬态解可以局部形式化地转化为（P_I）_I的二参数解。（T. Kawai和Y. Takei: Kokyuroku Bessatsu, B5 (2008)）每个（P_J）m （J= I， II， IV）同构于一个适当的退化Gamier系统，该系统被限制在某条复直线上。(T. Koike: Kokyuroku Bessatsu, B2 （2007), B5(2008)）除了对（P_J）m的研究取得进展外，我们还通过对另一种称为Noumi-Yamada系统的高阶painlevel方程的Stokes几何的研究，在虚拟拐点的研究方面取得了实质性进展。此外，T. Aoki、T. Kawai和Y. Takei （RIMS Preprint 1616(2007)）发现了利用无穷阶微分算子对精确WKB分析中若干无穷级数的新处理方法。围绕这些结果的会议于2008年1月28日至2月1日在法国马赛的CIRM （Centre International de Rencontres Mathematiques）举行。这是一次密集而富有成果的会议，有许多外国人（既不是法国人也不是日本人）参加。