Geometrical Models and their Applications

几何模型及其应用

• 批准号: 07640526
• 负责人: WADATI Miki
• 金额: $ 1.73万
• 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 (C)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640526/
• 关键词: Geometrical Model; Curve-Shortening-Equation; Curve-Lengthening Equation; Geometrical Phase; Discrete Integrable Equation; Motion of Curve; Motion of Surface; Integrable Equation; 離散化; 離散曲線の運動; 離散的可積分方程式; 離散的セレ・フレネ方程式; 結晶成長模型; 可積分模型; 離散的可積分模

Through a research on Geometrical Models and their Applications, the following results are obtained.1. Relation between motion of curve and integrable evolution equation is explained by an equivalence between Seret-Frenet equation and AKNs eigen-value problem at zero eigenvalue. This remains valid for discrete systems.2. Curve-lengthening equation is introduced and its exact solutions are found which include the Saffman-Taylor solution.3. Level-set formulation for motion of curve is introduced and the generalization of the Saffman-Taylor solution is derived.4. Time evolution of surface in a curved space is fomulated.5.Motion of triangularized surfaces in 3-dimensional space is formulated, and geometrical properties of discrete KdV equation and discrete Nonlinear Schrodinger equation are clarified.6. As a new type of geometrical models, a model specified by an acceleration field is propsed.Curve-shortening equation is discretized so as to retain its geometrical properties.The above results are useful not only in clarifyng mathematical structures of geometrical models but also in applying the models to various fields of physics.

通过对几何模型及其应用的研究，得到了以下结果。用零本征值下的Seret-Frenet方程与AKNS本征值问题的等价性来解释曲线运动与可积发展方程之间的关系。这对离散系统仍然有效。介绍了曲线加长方程，并求出了它的精确解，其中包括Saffman-Taylor解。介绍了曲线运动的水平集形式，推导了Saffman-Taylor解的推广。建立了曲面在三维空间中的运动公式，阐明了离散KdV方程和离散非线性薛定谔方程的几何性质。作为一种新的几何模型，提出了一种由加速度场表示的模型，对曲线缩短方程进行离散化，以保持其几何性质。上述结果不仅有助于阐明几何模型的数学结构，而且有助于将该模型应用于各种物理领域。

项目成果

K.Nakayama: "On the Level-Set Formulution of Geonmetrical Models" Journal of Physical Society of Japan. 64. 403-406 (1995)

K.Nakayama：“论几何模型的水平集公式”日本物理学会杂志。

K.Nakayama: "Reation-Diffusion Systemina Curved Space and the KPZ eguation" Journal of Physical Society of Japan. 64. 1501-1505 (1995)

K.Nakayama：“弯曲空间的反应扩散系统和 KPZ 方程”日本物理学会杂志。

M.Hisakado and M.Wadati: "Moving Discrete Curve and Geometrical Phase" Phys.Lerr.A214. 252-258 (1996)

M.Hisakado 和 M.Wadati：“移动离散曲线和几何相位”Phys.Lerr.A214。

M.Hisakado: "Integrable Dynamics of Discrete Surface II" Journal of Physical Society of Japan. 65. 389-393 (1996)

M.Hisakado：“离散表面的积分动力学II”日本物理学会杂志。

T.Tsurumi: "Motion of Curves Specified by Accelerations" Physics Letters A. 224. 253-263 (1997)

T.Tsurumi：“由加速度指定的曲线运动”《物理快报》A. 224. 253-263 (1997)