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方程的单调群。 （Kawai-Takei，奇异扰动的代数分析，第 3 章，Iwanami（日文））。（2）通过多尺度分析构造大参数 Painleve 方程的 2 参数形式解，然后将其显示出来形式上和局部地简化为 Painleve 方程 I 型的某个适当的 2 参数解。（Aoki-Kawai-Takei，在“Structure of Solutions of微分方程”，World Scientific 和 Kawai 和 Takei，Adv. in Math.，134)(3) 将哈密顿系统的奇异微扰简化为 Birkhoff 范式，这可以用作多尺度分析的更透明的替代品构造 Painleve 方程的 2 参数形式解。 （Takei，Publ. RIMS，34）（4）通过 Ansatz 介绍其 Stokes 几何，对具有大参数的高阶常微分方程进行精确 WKB 分析的试验。 （Aoki-Kawai-Takei，Asian J.Math. 2）（5）高阶非线性微分方程（如雅可比方程）解的自然边界的渐近分析。（6）非线性结构理论我们对(4)、(5)、(6)项的研究还处于初步阶段。

项目成果

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。

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。

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)。

河合隆裕: "特異摂動の代数解析学" 岩波書店(近刊予定), 140

Takahiro Kawai：“奇异扰动的代数分析”岩波书店（即将出版），140

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)