Theory of singular perturbations

(1)精确的WKB分析，即基于Borel求和的WKB分析，使我们能够用Borel恢复的WKB解的周期积分来描述二阶Fuchsian方程的单调群。(Kawai-Takei，奇异摄动的代数分析，第二章。(2)用多尺度分析方法构造了大参数Painleve方程的两参数形式解，并证明了它们的形式和局部归结为Painleve方程的某些适当的两参数形式解(World Science and Kawai and Takei，Adv.in Math，134)(World Science and Kawai and Takei，Adv.in Math，134)。(3)哈密顿系统的奇异摄动约化为Birkhoff范式，它可作为构造Painleve方程两参数形式解的更透明的多尺度分析的替代。(武井，Publ.RIMS，34)(4)通过Anatz关于其Stokes几何的介绍，尝试对具有大参数的高阶常微分方程作精确的WKB分析。(Aoki-Kawai-Takei，亚洲数学J.2)(5)高阶非线性微分方程解的自然边界的渐近分析(如Jacobi方程)。(6)Painleve方程以外的非线性方程的结构理论。我们对第(4)、(5)和(6)项的研究仍处于初步阶段。

(1) Exact WKB analysis, i.e., WKB analysis based on the Borel summation has enabled us to describe the monodromy group for second order Fuchsian equations in terms of period integrals of Borel resummed WKB solutions. (Kawai-Takei, Algebraic Analysis of Singular Perturbations, Chap. 3, Iwanami (in Japanese)).(2) 2-parameter formal solutions of Painleve equations with a large parameter are constructed by multiple-scale analysis, and then they are shown to be formally and locally reduced to some appropriate 2-parameter solution of the Painleve equation, type I.(Aoki-Kawai-Takei, in "Structure of Solutions of Differential Equations", World Scientific and Kawai and Takei, Adv. in Math., 134)(3) Singular-perturbative reduction of a Hamiltonian system to the Birkhoff normal form, which may be used as a more transparent substitute of multiple-scale analysis in constructing 2-parameter formal solutions of Painleve equations. (Takei, Publ. RIMS, 34)(4) A trial of exact WKB analysis for higher order ordinary differential equations with a large parameter through the presentation of Ansatz concerning their Stokes geometry. (Aoki-Kawai-Takei, Asian J.Math. 2)(5) Asymptotic analysis of natural boundaries of solutions of non-linear differential equations of higher order (such as the Jacobi equation).(6) Structure theory for non-linear equations other than Painleve equations.Our study of items (4), (5) and (6) still remain on a preliminary stage.