Geometrical Models and their Applications

通过对几何模型及其应用的研究，得到了以下结论.利用Seret-Frenet方程与AKNs特征值问题在零特征值处的等价性，说明了曲线的运动与可积发展方程之间的关系。这仍然适用于离散系统。2.引入曲线延长方程，并求出其精确解，其中包括Saffman-Taylor解.引入了曲线运动的水平集公式，并导出了Saffman-Taylor解的推广. 5.建立了三维空间中三角化曲面的运动方程，阐明了离散KdV方程和离散非线性Schrodinger方程的几何性质.本文提出了一种新的几何模型--加速度场模型，并对曲线缩短方程进行了离散化，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，而且有助于将几何模型应用于物理学的各个领域。

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.