Theory and Applications of Nonlinear Dynamical Systems

非线性动力系统理论与应用

基本信息

批准号：
06044054
负责人：
WADATI Miki
金额：
$ 5.38万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for international Scientific Research
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06044054/
关键词：
Soliton Integrable system Inverse scattering method Yangian Geometrical model Quanturi integrable system Nonlinear wave Discrete integrable system Nonlinear Schrodinger equation 光ソリトン W-代数カロジェロ系ヤン・バクスター関係式面の運動

项目摘要

Theory and Applications of Nonlinear Dynamical SystemsIn this project.a main theme is the analysis of nonlinear dynamical systems with large degrees of freedom which appear in various fields of physics. The followings are summary of the results.1. Quantum integrable systems with long-range interactionsWe have developped the quantum inverse scattering method for onedimensional quantum particle systems with long-range interactions. Integrabilities of the Calogero-Moser model and the Sutherland model are proved. Those models are extended so as to include internal degrees of freedom (spins). The Dunkl operator approach and the exchange operator approach are also clarified.2. Geometrical modelsThe purposes are two folds. One is the extension of the soliton systems to higher-dimensional ones and the other is a general setting for descriptions of geometrical objects in physics. We have developped the level-set formulation of the surfaces in arbitrary space-dimension. Special solutions of the curve-lengthening equation which generalize the Saffman-Taylor finger solution are found.3. Discrete dynamical modelsThe geometrical models are extended into the discrete curves and the discrete surfaces. The relations with descrete integrable systems (the discrete Modified K-dV hierarchy) are found.4. Two-dimensional integrable systemsThe Davey-Stewartson equation which is considered to be a two-dimensional extension of the nonlinear Schrodinger equation is studied numerically. The stability of dromions and the role of the mean flows are clarified.5. Random KnottingAs a model for polymers, topological configurations of random walks are investigated. Probability of a knot K as a function of the length N.P (K.N).is determined by numerical experiments. The proposed formula for B (K.N) agrees well with the numerical results.

非线性动力学系统在这个项目中的理论和应用。主要主题是对具有较大自由度的非线性动力学系统的分析，这些系统出现在各个物理学领域。以下是结果的摘要1。具有远距离相互作用的量子整合系统我们已经开发了具有远距离相互作用的一度量子粒子系统的量子反向散射方法。证明了Calogero-Moser模型和Sutherland模型的集成能力。这些模型被扩展，以包括内部自由度（旋转）。还阐明了DUNKL操作员方法和交换运算符方法2。几何模型的目的是两个倍。一个是将孤子系统扩展到较高维度的系统，另一个是描述物理学几何对象的一般环境。我们已经在任意空间维度中开发了表面的级别公式。发现了曲线延长方程的特殊溶液，从而概括了Saffman-Taylor指的溶液3。离散动力学模型将几何模型扩展到离散曲线和离散表面。发现与解描述集成系统（离散修改的K-DV层次结构）的关系4。二维集成系统的Davey-Stewartson方程被认为是非线性Schrodinger方程的二维扩展。阐明了Dromions的稳定性和平均流的作用。5。随机打结一个聚合物的模型，研究了随机步行的拓扑结构。结k的概率是长度N.P（K.N）的函数。由数值实验确定。 B（K.N）提出的公式与数值结果非常吻合。