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

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，并给出可分析的非线性离散模型，我们得到了以下结果：(1)研究了三维空间中可扩展弦的离散模型，提出了用孤子理论分析空间中弦的新方法。我们利用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，并成为特殊的Cosserat弹性弦的连续极限。(2)建立了准各向同性增强随机斩波玻璃/聚丙烯复合材料的静强度模型。该模型将宏观量和微观量联系起来，有望用于多体分析。(3)研究了多体动力学离散模型的离散孤子方程，证明了可扩展弦的运动可以用双分量KP方程的孤子解很好地描述。(4)对反应扩散系统和交通流等离散系统进行了研究，结果表明，通过适当的离散化处理，可以提取出离散现象的本质部分。

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.