Structural analysis of differential equations by the exact WKB method

通过精确 WKB 方法进行微分方程的结构分析

• 批准号: 14340042
• 负责人: KAWAI Takahiro
• 金额: $ 4.22万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340042/
• 关键词: exact WKB analysis; a higher order Painleve equation; The Toulouse Project; a microdifferential operator of WKB type; a virtual turning point; the exact steepest descent method; singular perturbation theory; algebraic analysis; 高階パンルヴェ方式; Toulous

1.A structure theorem for solutions of a higher order Painleve equation.Kawai and Takei(Proc. Japan Acad., 80A(2004) have succeeded in transforming any 0-parameter solution of any member of the Painleve hierarchy(P_J)(J=I, II-1,II-2) to that of the classical Painleve-I equation near its turning point of the first kind. This success has made us convinced that the exact WKB analysis should be efficient in studying the structure of higher order Painleve equations, leading to our Announcement of the Toulouse Project(RIMS Kokyuroku 1397(2004)).2.Proposal of the exact steepest descent method, and its applications. We have confirmed the efficiency of the newly proposed method, the exact steepest descent method, in the situation where the traditional WKB method cannot be applied. (Aoki-Kawai-Takei : Adv. Studies in Pure Math., 42(2004) ; Aoki-Koike-Takei, and Koike-Takei : in Microlocal Analysis and Complex Fourier Analysis, World Scientific,2002).3.Motivated by a basic integral equation in the plasma physics, we have developed the exact WKB analysis for (micro)differential operators of WKB type. (Aoki-Kawai-Koike-Takei : Adv.in Math.,181(2004),Ann. Inst. Fourier,54 (2004), Contemporary Math.,373(2005), AMS).4.We have made it manifest that the notion of virtual turning points introduced by Aoki-Kawai-Takei is an indispensable one in the WKB analysis of higher order differential equations. (Kawai-Koike-Nishikawa-Takei : Asterisque,297(2004), Soc. Math. France ; Aoki-Kawai-Sasaki-Shudo-Takei : J.Phys., A38(2005)).

1.高阶 Painleve 方程解的结构定理。Kawai 和 Takei(Proc. Japan Acad., 80A(2004) 成功地变换了 Painleve 层次结构中任意成员的任意 0 参数解(P_J)(J= I, II-1,II-2) 到经典 Painleve-I 方程接近其第一类转折点这一成功使我们确信 WKB 分析是精确的。在研究高阶 Painleve 方程的结构时应该是有效的，导致我们宣布了图卢兹项目（RIMS Kokyuroku 1397（2004））。2.精确最速下降法的建议及其应用我们已经证实了它的效率。新提出的方法，即精确最速下降法，在传统 WKB 方法无法应用的情况下（Aoki-Kawai-Takei：Adv. Studies in Pure Math.， 42(2004) ; Aoki-Koike-Takei, and Koike-Takei : in Microlocal Analysis and Complex Fourier Analysis, World Scientific,2002).3.受等离子体物理学中基本积分方程的启发，我们开发了精确的 WKB 分析对于 WKB 类型的（微）微分运算符。 (Aoki-Kawai-Koike-Takei : Adv.in Math.,181(2004),Ann. Inst. Fourier,54 (2004), Contemporary Math.,373(2005), AMS).4.我们已经表明了这一点Aoki-Kawai-Takei 提出的虚拟转折点的概念是高阶微分方程 WKB 分析中不可或缺的一个概念。 (Kawai-Koike-Nishikawa-Takei：Asterisque，297(2004)，法国数学学会；Aoki-Kawai-Sasaki-Shudo-Takei：J.Phys.，A38(2005))。

项目成果

On WKB analysis of higher order Painleve equations with a large parameter.

大参数高阶 Painleve 方程的 WKB 分析

• 发表时间: 2004
• 期刊: Proc. Japan Acad., Ser. A. 80
• 作者: NASIR;Haniffa Mohamed;KAKO;Takashi;KOYAMA;Daisuke;Masato TSUJII;H.mizuno;T.Kawai
• 通讯作者: T.Kawai

Takahiro Kawai, T.Aoki, Y.Takei: "The exact steepest descent method - A new steepest descent method based on the exact WKB analysis"Adv. in Pure Math.. (to appear).

Takahiro Kawai、T.Aoki、Y.Takei：“精确最速下降法 - 基于精确 WKB 分析的新最速下降法”Adv.

[KS] T.Kawai, H.P.Stapp: "On infra-red singularities associated with QC photons"Microlocal Analysis and Complex Fourier Analysis, World Scientific. 115-134 (2002)

[KS] T.Kawai、H.P.Stapp：“与 QC 光子相关的红外奇点”微局域分析和复杂傅立叶分析，世界科学。

[AKoT]T.Aoki, T.Koike, Y.Takei: "Vanishing of Stokes curves"Microlocal Analysis and Complex Fourier Analysis, World Scientific. 1-22 (2002)

[AKoT]T.Aoki、T.Koike、Y.Takei：“斯托克斯曲线的消失”微局域分析和复杂傅立叶分析，世界科学。

[AKY]青木貴史, 片岡清臣, 山崎晋: "超函数・FBI変換・無限階擬微分作用素"共立出版 近刊. (2003)

[AKY] Takashi Aoki、Kiyoomi Kataoka、Susumu Yamazaki：“变换函数、FBI 转换和无限阶伪微分算子”Kyoritsu Shuppan，即将出版 (2003)。