Analysis and Control of Multi-body Dynamics by Nonlinear Discrete Integrable Theory

非线性离散可积理论的多体动力学分析与控制

• 批准号: 10554002
• 负责人: SATSUMA Junkichi
• 金额: $ 8.9万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10554002/
• 关键词: Soliton; Extensible String; Discrete Soliton; Integrable System; KP方程式

For the purpose of classifying nonlinear discrete equations and their approximations in the series of fundamental equations appearing in multi-body dynamics, and presenting analyzable nonlinear discrete model, we have a obtained the following results.(1) We have studied a discrete model of an extensible string in three dimensional space and presented a new method of analyzing a string in space by the soliton theory. We have obtained some exact solutions by the soliton theory. The discrete basic equations are suitable for numerical simulations of string dynamics. Moreover, The model contains the bending and twisting, and becomes the special Cosserat elastic string art the continuous limit.(2) We have proposed a model to study the static strength of a quasi-isotropically reinforced random chopping glass/polypropylene composite. The model relates macroscopic and microscopic quantities, and is expected to be useful for the multi-body analysis.(3) We have studied discrete soliton equations which are reductions of the discrete model of multi-body dynamics and shown that the motion of a extensible string is well described by a soliton solution of two-component KP equation.(4) We have studied several discrete systems such as reaction-diffusion system and traffic flow, and shown that essential part of the phenomena are extracted by suitable discretizations.

为了对多体动力学中出现的一系列基本方程中的非线性离散方程及其近似进行分类，并提出可解析的非线性离散模型，我们得到了以下结果：(1)研究了多体动力学的离散模型提出了一种利用孤子理论分析空间弦的新方法。我们通过孤子理论得到了一些精确解。离散基本方程适用于弦动力学的数值模拟。此外，该模型包含了弯曲和扭转，成为特殊的Cosserat弹力弦艺术的连续极限。(2)提出了一种研究准各向同性增强随机短切玻璃/聚丙烯复合材料静态强度的模型。该模型将宏观和微观的量联系起来，有望对多体分析有用。(3)我们研究了离散孤子方程，它是多体动力学离散模型的简化，并表明可扩展的运动弦可以用二分量KP方程的孤子解很好地描述。(4)我们研究了几个离散系统，如反应扩散系统和交通流，并表明现象的本质部分可以通过适当的离散化来提取。

项目成果

M.J.Ablowitz: "On Discretization of the Vector Nonlinear Schrodinger Equation"Phys. Lett. A. 253. 287-304 (1999)

M.J.Ablowitz：“关于向量非线性薛定谔方程的离散化”Phys。

K. Nishinari: "A Lagrange Representation of Cellular Automaton Models of Traffic Flow"J. Phys. A : Math. Gen.. Vol. 34. 10727-10736 (2001)

K. Nishinari：“交通流元胞自动机模型的拉格朗日表示”J。

A. Nobe: "Stable Difference Equations Associated with Elementary Cellular Automata"Japan J. Indust. Appl. Math.. Vol. 18, N0. 2. 293-305 (2001)

A. Nobe：“与基本元胞自动机相关的稳定差分方程”日本工业杂志。

T. Tokihiro: "Proof of Solitonical Nature of Box and Ball System by means of Inverse ultra-discretization"Inverse Problem. Vol. 15. 1639-1662 (1999)

T. Tokihiro：“通过逆超离散化证明盒球系统的孤子性质”逆问题。

A. Nagai: "Conserved Quantities of Box and Ball System"Glasgrow Math. J.. Vol. 43A. 91-97 (2001)

A. Nagai：“盒子和球系统的守恒量”格拉斯格罗数学。