Studies of Nonlinear Waves and Nonlinear Dynamical Systems

1.对于一维XXZ链这一已知的量子可积自旋系统，计算了非对称情况下的标量积和相关函数，以及有界情况下的自发磁化强度.研究了磁阱下玻色-爱因斯坦凝聚体的静态和动态性质。详细分析了D维非线性薛定谔方程的稳定性，二分量玻色子系统的稳定性，玻色子-费米子系统的动力学，以及各向异性凝聚体的基态及其稳定性.得到了描述下落细丝的Navier-Stokes方程的精确解，并对解进行了线性稳定性分析，给出了端点和中间点夹断的判据. Calogero模型、Sutherland模型和Ruijsenaars模型被称为量子可积粒子系统。系统地阐明了它们的代数结构、可积性和正交基.通过推广逆散射方法，得到了离散的多分量孤子方程及其解。得到了一类新的离散多分量非线性薛定谔方程.对于沃尔泰拉方程和Bogoyavlensky格，阐明了代数结构和可积性。进一步，通过离散时间和因变量，构造了可积细胞自动机.本文发展了一个处理量子可积粒子系统的费米子R矩阵理论。这一发展使我们能够研究费米子系统，而不求助于Jordin-Wigner变换。可积边值问题也可以处理。

1. For one-dimensional XXZ chain which is known to be quantum integrable spin system, scalar products and correlation functions of the asymmetric case, and spontaneous magnetization of the bounded case are calculated explicitly.2. Static and dynamical properties of Bose-Einstein condensate under magnetic trap are investigated. Stability of D-dimensional nonlinear Schrodinger equation, stability of 2-component boson system, dynamics of boson-fermion system, and ground state and its stability of anisotropic condensate are analysed in detail.3. An exact solution of the Navier-Stokes equations which describes a falling filament is found. The linear stability analysis of the solution gives a criterion for the pinch-off at the end points and the intermediate points.4. Calogero model, Sutherland model and Ruijsenaars model are known as quantum integrable Particle systems. Their algebraic structures, integrabilities and orthogonal bases are clarified in a systematic way.5. By extending the inverse scattering method, discrete multi-component soliton equations and their solutions are obtained. A new type of discrete multi-component nonlinear Schrodinger equation is also obtained.6. For Volterra equation and Bogoyavlensky lattice, algebraic structures and integrabilities are clarified. Further, by discretizing time and dependent variables, integrable cellular automata are constructed.7. A theory of fermionic R-matrix is developed to treat quantum integrable particle systems. This development enables us to study fermion systems without recourse to the Jordin-Wigner transformation. The integrable boundary problem can be treated as well.