New Development of Singular-Perturbation Method with Renormalization Group

重正化群奇异摄动方法的新进展

• 批准号: 13640402
• 负责人: NOZAKI Kazuhiro
• 金额: $ 1.54万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640402/
• 关键词: Singular perturbation method; renormalization group; symplectic map; Lie symmetry; phase equation; similarity-type solution; Lie symmetry; 非線形偏微分方程式; 不変解; シンプレクティク写像; ポアンカレーバーコフ分岐; くり込み群の方法; symplectic map; reduced map; 共鳴島形成; Liouville exp

The long-time asymptotic behavior of a nonlinear dynamical system has been studied by various singular perturbation methods, such as the multi-time method, the normal form theory and the reductive perturbation method. Those various perturbation methods have been unified by the recent development of the perturbative renormalization group method (the RG method), where the long-time asymptotic behavior of a system is described by the renormalization group equation. In this work, the RG method is developed as follows.1.A symplecticity-preserving RG method is formulated and is applied to the Poicare-Birkhoff bifurcation of a two-dimensional symplectic map. We obtain analytical expressions to the resonant island structure.2.A proto-RG approach of the RG method is proposed to avoid the step of obtaining explicit secular solutions in the RG method. Various phase equations are systematically derived as RG equations from a general reaction-diffusion equation.3.The RG method with the Lie symmetry is extended to apply to interesting physical systems such as a gas sphere under gravity and adiavatic perfect gas, which have only trivial Lie symmetries.

利用各种奇异摄动方法，如多时间法、范式理论和约化摄动方法，研究了非线性动力系统的长期渐近行为。这些不同的微扰方法已经被最近发展的微扰重整化群方法（RG方法）统一起来，其中系统的长时间渐近行为由重整化群方程描述。在这项工作中，RG方法的发展如下。提出了一种保辛RG方法，并将其应用于二维辛映射的Poicare-Birkhoff分岔问题。我们得到了共振岛结构的解析表达式。提出了一种RG方法的原型RG方法，避免了RG方法中获取显式长期解的步骤。从一般的反应扩散方程出发，系统地推导出各种相方程为RG方程。将具有李对称的RG方法推广到一些有趣的物理系统，如重力作用下的气体球和绝热完美气体，它们只有平凡的李对称。

项目成果

S.Goto, K.Nozaki: "Liouville Operator Approach to Symplecticity-Preserving RGM"Physica D. (in press). (2004)

S.Goto、K.Nozaki：“Liouville 算子方法保辛性 RGM”Physica D.（出版中）。

Regularized Renormalization Group Reduction of Symplectic Maps

辛映射的正则重整化群约简

• 发表时间: 2001
• 期刊: J.Phys.Soc.Jpn. 70(1)
• 作者: S.Goto;K.Nozaki
• 通讯作者: K.Nozaki

Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map

二维辛图中共振结构的重整化分析

• 发表时间: 2004
• 期刊: Prog.Theor.Phys. 111(4)
• 作者: S.Murata;S.Goto;T.Maruo
• 通讯作者: T.Maruo

Dynamics of Gas Sphere under Self-Gravity.

自重力作用下的气体球动力学。

• 发表时间: 2004
• 期刊: Phys.Letters A332
• 作者: S.Murata;S.Goto;T.Maruo;S.Murata
• 通讯作者: S.Murata

A Phase Equation of Third-Order Spatial Derivative

三阶空间导数的相位方程

• 发表时间: 2002
• 期刊: Prog.Theor.Phys. 107(2)
• 作者: S.Murata;K.Nozaki;Y.Masutomi
• 通讯作者: Y.Masutomi