Theory of singular perturbations

奇异摄动理论

• 批准号: 08454029
• 负责人: KAWAI Takahiro
• 金额: $ 2.37万
• 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: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454029/
• 关键词: Exact WKB analysis; Borel resummation; monodromy groups; Painleve transcendents; multiple-scale; Schrodinger equations; Stokes curves; deformation (of differential equations); 接続公式; 変わり点; (微分方程式の)変形; パンルヴェ函数; multiple-scale; ストークス曲線; 変り点; ボ

(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.

(1)精确WKB分析，即基于Borel求和的WKB分析，使我们能够用Borel恢复WKB解的周期积分来描述二阶Fuchsian方程的单群。（Kawai-Takei，奇异摄动的代数分析，第三章，Iwanami）。(2)通过多尺度分析构造了大参数painlevel方程的2参数形式解，并证明了它们的形式和局部约简为painlevel方程的i型适当的2参数解（Aoki-Kawai-Takei，“Structure of solutions of Differential equations”，World Scientific and Kawai and Takei, ad . in Math）。(3) hamilton系统的奇异摄动约简为Birkhoff范式，在构造painlevel方程的2参数形式解时，它可以作为多尺度分析的一个更透明的替代。(武井,出版。(4)利用关于Stokes几何的Ansatz的描述，对具有大参数的高阶常微分方程的精确WKB分析进行了尝试。（Aoki-Kawai-Takei, Asian J.Math）(5)高阶非线性微分方程（如Jacobi方程）解的自然边界的渐近分析。(6) painlevel方程以外的非线性方程的结构理论。我们对第(4)、(5)、(6)项的研究仍处于初步阶段。

项目成果

Y.Takei: "Singular-perturbative reduction to Birkhoff normal form and instanton-type formal solutions of Hamiltonian systems" Publ.RIMS,Kyoto Univ.34. 601-627 (1998)

Y.Takei：“哈密顿系统的 Birkhoff 范式和瞬子型形式解的奇异微扰还原”Publ.RIMS，Kyoto Univ.34。

T.kawai, Y.Takei: "WKB analysis of Painleve transcendents with a large parameter.III-Local reduction of 2-parameter Painleve transcendents" Adv.in Math.134. 178-218 (1998)

T.kawai、Y.Takei：“具有大参数的 Painleve 超越项的 WKB 分析。III-2 参数 Painleve 超越项的局部简化”Adv.in Math.134。

河合隆裕: "WKB analysis of Painleve transcendents with a large parameter. II -- Multiple-scale analysis of Painleve transcendents. --" Structure of Solutions of Differential Equations, World Scientific. 1-49 (1996)

Takahiro Kawai：“具有大参数的 Painleve 超越项的 WKB 分析。II -- Painleve 超越项的多尺度分析。--”微分方程解的结构，世界科学 1-49 (1996)。

岡本久: "関数解析1,2" 岩波書店, 266 (1997)

冈本恒：《泛函分析 1,2》岩波书店，266 (1997)

伊原康隆: "On deformation of maximally degenerate stable marked curves and Oda's problem" J.reine angew.Math.(in press).

Yasutaka Ihara：“关于最大退化稳定标记曲线的变形和 Oda 问题”J.reine angew.Math.（出版中）。