Theory and Applications of Nonlinear Dynamical Systems

非线性动力系统的理论与应用在这个项目中，一个主要的主题是分析在物理学的各个领域中出现的具有大自由度的非线性动力系统。以下是结果的总结。具有长程相互作用的量子可积系统我们发展了具有长程相互作用的一维量子粒子系统的量子逆散射方法。证明了Calogero-Moser模型和Sutherland模型的可积性。这些模型进行了扩展，以包括内部自由度（自旋）。对Dunkl算子方法和交换算子方法也进行了阐述.几何模型的目的有两个方面。一个是孤子系统向高维系统的扩展，另一个是物理学中描述几何对象的一般设置。我们建立了任意空间维曲面的水平集公式。得到了曲线延长方程的特解，推广了Saffman-Taylor指形解.离散动力学模型将几何模型推广到离散曲线和离散曲面。建立了与离散可积系统（离散修正K-dV族）的关系.二维可积系统数值研究了Davey-Stewartson方程，它被认为是非线性Schrodinger方程的二维推广。阐明了dromion的稳定性和平均流的作用.作为聚合物的一种模型，研究了无规行走的拓扑构型。通过数值实验确定了纽结概率K与长度N.P（K.N）的函数关系。所提出的B（K.N）计算公式与数值计算结果吻合较好。

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.