Studies of Nonlinear Waves and Nonlinear Dynamical Systems

非线性波和非线性动力系统的研究

• 批准号: 09044065
• 负责人: WADATI Miki
• 金额: $ 7.1万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09044065/
• 关键词: Soliton; Nonlinear dynamics; Discrete integrable system; Integrable system; Bose-Einstein Condenisation; Quantum integrable system; Instability; Geometrical model; 流体フィラメントの自由落下; ブラック格子; ギャップ・ソリトン; フェルミオン的R行列; ソリトン方程式; 離散的ソリトン方程式; ソリトン・セルラーオ

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.

1. 对于已知为量子可积自旋系统的一维XXZ链，明确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。研究了磁阱作用下玻色-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、双组分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态的基态及其稳定性。找到了描述落丝的Navier-Stokes方程的精确解。对解的线性稳定性分析给出了端点和中间点掐断的判据。Calogero模型、Sutherland模型和rujsenaars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，得到离散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Volterra方程和Bogoyavlensky格，澄清了代数结构和可积性。进一步，通过离散时间和因变量，构造了可积元胞自动机。提出了一种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需要借助约丁-维格纳变换。可积边界问题也可以处理。

项目成果

H. Morise: "Stability of a Two-Component Bose-Einstein Condensute"Journal of Physical Society of Japan. 68. 1871-1876 (1999)

H. Morise：“二元玻色-爱因斯坦凝聚体的稳定性”日本物理学会杂志。

A. Nishino et al.: "Rodrigues Formula for the Nonsymmetric Madconald Polynomial"Journal of Physical Society of Japan. 68. 701-704 (1999)

A. Nishino 等人：“非对称 Madconald 多项式的罗德里格斯公式”日本物理学会杂志。

T. Tsuchida: "New Integrable systems of derivative nonlinear Schrodinger equations with multiple components"Physics Letters A. 257. 53-64 (1999)

T. Tsuchida：“具有多个分量的导数非线性薛定谔方程的新可积系统”《物理快报》A. 257. 53-64 (1999)

H.Ujino: "Algebraic Construction of a New Symmetric Orthogonal Basis for the Calogero Model" J.Phys.Soc.Jpn.67. 1-4 (1998)

H.Ujino：“Calogero 模型的新对称正交基的代数构造”J.Phys.Soc.Jpn.67。

A.Nishino: "Symmetric Fock Space and Orthogonal Symmetric Polynomials Associated with the Calogero Model" Chaos,Solitons & Fractals. (印刷中).

A.Nishino：“与 Calogero 模型相关的对称福克空间和正交对称多项式”混沌、孤立子和分形（正在出版）。