Structural analysis of differential equations by the exact WKB method

1.高阶Painleve方程解的结构定理.Kawai和Takei(Proc.日本学报，80A(2004)已成功地将Painleve方程(P_J)(J=I，II-1，II-2)的任意零参数解转化为第一类转折点附近的经典Painleve-I方程的解。这一成功使我们相信，精确的WKB分析在研究高阶Painleve方程的结构方面应该是有效的，从而导致我们宣布了图卢兹计划(RIMS Kokyuroku 1397(2004))。2.精确最陡下降法的提出及其应用。在传统WKB方法不能应用的情况下，我们证实了新提出的方法--精确最陡下降法的有效性。(Aoki-Kawai-Takei：高级纯数学研究，42(2004)；Aoki-Koike-Takei，and Koike-Takei：in Microlocal Analyses and Complex傅立叶分析，World Science，2002)。3.在等离子体物理中一个基本积分方程的推动下，我们发展了WKB型(微)微分算子的精确WKB分析。青木-河井-小池-武井：数学进修，181(2004)，安。安装傅立叶，54(2004)，当代数学，373(2005)，AMS)。4.我们已经表明，由Aoki-Kawai-Takei引入的虚拟转折点的概念是高阶微分方程WKB分析中不可缺少的概念。(Kawai-Koike-西川-Takei：Asterisque，297(2004)，Soc.数学课。法国；青木-川井-佐佐木-舒道-武井：《物理杂志》，A38(2005))。

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)).