1997 Fiscal Year Final Research Report Summary
Geometrical Models and their Applications
| Project/Area Number |
07640526
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
物理学一般
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| Research Institution | The University of Tokyo |
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Principal Investigator |
WADATI Miki The University of Tokyo, Graduate School of Science, Professor, 大学院・理学系研究科, 教授 (60015831)
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| Project Period (FY) |
1995 – 1997
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| Keywords | Geometrical Model / Curve-Shortening-Equation / Curve-Lengthening Equation / Geometrical Phase / Discrete Integrable Equation / Motion of Curve / Motion of Surface / Integrable Equation |
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| Research Abstract |
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.
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