Structural analysis of differential equations by the exact WKB method

1.一类高阶painlevel方程解的结构定理。Kawai和Takei（续）日本Acad., 80A（2004）成功地将painlevel层次（P_J）（J=I, II-1,II-2）中任意成员的任意0参数解转化为经典painlevel -I方程在第一类拐点附近的解。这一成功使我们确信精确的WKB分析在研究高阶painlevel方程的结构方面应该是有效的，这导致了我们的图卢兹项目公告（RIMS Kokyuroku 1397(2004)）。精确最陡下降法的提出及其应用。在传统WKB方法无法应用的情况下，我们证实了新提出的方法——精确最陡下降法的有效性。青木川井武井：《纯数学研究》， 42(2004)；2 .青木小池武井和小池武井：微局部分析和复傅立叶分析，世界科学，2002)。基于等离子体物理学中的一个基本积分方程，我们对WKB型（微）微分算子进行了精确的WKB分析。（青木川井小池武井：数学学报，181(2004)，安。）傅立叶研究所，54(2004)，当代数学。373 (2005), AMS)。4。在高阶微分方程的WKB分析中，由Aoki-Kawai-Takei引入的虚拟拐点的概念是不可缺少的。（川井小池西川武井：Asterisque，297(2004), Soc.）数学。法国;青木川井佐佐木修道武井：j.s phys。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)).