Study of Diophantine Phenomena Appearing in Asymptotic Analysis of Nonlinear Partial Differential Equations

非线性偏微分方程渐近分析中的丢番图现象研究

• 批准号: 11640183
• 负责人: YOSHINO Masafumi
• 金额: $ 1.54万
• 依托单位: CHUO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640183/
• 关键词: nonlinear Fuchsian system; Diophantine condition; resonance; Frobenius theorem; singular solution; asymptotic solution; simultaneous normal forms; WKB method; ベクトル場の標準形; ハミルトニアンの標準形; diophantine条件; 連分数; 積分可能性; 同時diophantin条件; 積分可能; 同時ポアンカレ条

We have obtained two main results 1) and 2).1) We reveal the role of Diophantine conditions when solving a system of nonlinear partial differential equations or difference equations in the normal form theory. Namely we havea) The simultaneous normal form of a commuting system of maps under Bruno condition is presented. This is also valid for a system of circle maps in a Gevrey class.b) The solvability of nonlinear Fuchsian equations of general variables in a class of finitely smooth functions is proved. This yields a so-called Grobman-Hartmann type theorem.c) The global normal form of a pseudodifferential operator on tori which is a perturbation of a system of constant vector fields is presented. Necessary and sufficient conditions are given in terms of a Diophantine condition.d) To make clear the relations between a simultaneous Diophantine condition of a commuting system of maps or vector fields and the Diophantine conditions for every generator of the system.2) We show that a Riemann-Hilbert problem is closely related to the solvability of equations without Diophantine conditions. Namely we havea) Solvability of nonlinear Fuchsian equations of general variables under a so-called Riemann-Hilbert condition is proved. This is applied to a mixed Monge- Ampere equation.b) Solvability of the same equations as in a) in a class of singular functions under a Poincare condition is presented. This can be regarded as a Frobenius theorem for a partial differential equation. The results are partly reported in ICM conference in Beijing in 2002.

我们已经获得了两个主要结果1）和2）.1）我们在求解正常形式理论中非线性偏微分方程或差异方程的系统时揭示了双苯胺条件的作用。即，在布鲁诺条件下，我们没有同时的地图通勤系统的正常形式。这对于在Gevrey类中的圆形图系统也有效。B）证明了一类有限平滑函数中一般变量的非线性紫红色方程的溶解性。这产生了所谓的grobman-hartmann型定理。 d）为了清楚地图的通勤系统或矢量场的通勤系统与系统的每个发生器的养分状况之间的同时养分状况之间的关系。也就是说，在所谓的riemann-hilbert条件下，非线性变量的非线性紫红色方程已被证明是可以的。 b）与a）在庞加罗条件下的一类奇异函数中相同的方程式的求解性。这可以被视为偏微分方程的frobenius定理。结果在2002年在北京举行的ICM会议上部分报告。

项目成果

Masafumi Yoshino: "Simultaneous normal forms of commuting maps"Proc. SPT98 (World Scientific). (2000)

Masafumi Yoshino：“通勤地图的同时规范形式”Proc。

Masafumi Yoshino: "Todor Gramchev; Rapidly convergent iteration method for simultaneous normal forms of commuting maps"Math. Z.. 231. 745-770 (1999)

Masafumi Yoshino：“Todor Gramchev；通勤地图同时范式的快速收敛迭代方法”数学。

Masafumi Yoshino: "Singular Solutions of Nonlinear Fuchsian Equations and Applications to Normal Form Theory"数理研講究録. 1296. 73-79 (2002)

Masafumi Yoshino：“非线性 Fuchsian 方程的奇异解及其在范式理论中的应用”数学研究报告 1296. 73-79 (2002)。

Masafumi Yoshino: "Simultaneous normal forms of commuting maps"Proc. SPT98 (World Scientific). 287-294 (2000)

Masafumi Yoshino：“通勤地图的同时规范形式”Proc。

Masafumi Yoshino, Todor Gramchev, Detta Dickinson: "Perturbations of vector fields on tori : resonant normal forms and Diophantine phenomena"Proc. Edinburgh Math. Soc.. 45. 731-759 (2002)

Masafumi Yoshino、Todor Gramchev、Detta Dickinson：“环面矢量场的扰动：共振范式和丢番图现象”Proc。