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.较高阶段方程的解决方案的结构定理。卡瓦伊和takei（日本学院，80a（2004年），已经成功地改变了对潘勒夫层次结构（p_j）的任何0个参数的解决方案（j = i = i，ii，ii-1，ii-2），应该在经过的painleve-ii equartion近期进行何处。有效地研究高阶次要方程式的结构，导致我们宣布了Toulouse项目（RIMS Kokyuroku 1397（2004））。2。确切的最陡峭下降方法的实用性，我们的应用已确认新提出的方法的效率。 Pure Math。，42（2004）; （Aoki-Kawai-Koike-Takei：Adv.in Math。，181（2004），Ann。Inst。Fourier，54（2004），《现代数学》，373（2005），AMS）.4。我们已经表明，Aoki-Kawai-takei的虚拟转折点的概念是Aoki-Kawai-takei的概述，这是一个较高的秩序分析。 （Kawai-Koike-Nishikawa-Takei：Asterisque，297（2004），Soc。Math。France; Aoki-Kawai-Sasaki-Sasaki-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.

[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。

[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。

[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)。