2003 Fiscal Year Final Research Report Summary
Research on algebraic structures of ultradiscrete soliton equations and applications to engineering
| Project/Area Number |
13640212
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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 |
Global analysis
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| Research Institution | Osaka University |
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Principal Investigator |
NAGAI Atsushi Nihon University, College of Industrial Technology, Lecturer, 基礎工学研究科, 助手 (90304039)
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| Co-Investigator(Kenkyū-buntansha) |
KONDO Koichi Doshisha University, Department of Engineering, Lecturer, 工学部, 講師 (30314397)
TSUJIMOTO Satoshi Kyoto University, Graduate School of Informatics, Lecturer, 大学院・情報学研究科, 講師 (60287977)
OKADO Masato Osaka University, Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (70221843)
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| Project Period (FY) |
2001 – 2003
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| Keywords | box and ball systems / integrable system / discrete, ultradiscrete / fractional derivative / MIttag-Leffler function / Green function / inverse ultradiscretization / RI chain |
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| Research Abstract |
The purpose of this research is to find algebraic and analytical structures of differential, difference and ultradiscrete equations, and to apply, the mathematical results to the field of engineering. The main results obtained are as follows.
1. We have found conserved quantities of box and ball systems with arbitrary box capacities by making use of ultradiscrete KdV and Lotka-Volterra equation including their modified version.
2. We have obtained piecewise linear equation of combinatorial R of crystals associated with D type affine Lie algebra by employing inverse ultra-discretization.
3. A discrete integrable system called the RI chain, which is considered as a generalization of the Toda equation, is studied. In particular, we have obtained its bilinear form and its determinant solution. The relation with relativistic Toda equation is also found.
4. We have obtained discrete and q-discrete versions of fractional derivative together with ite eigen function called the Mittag-Leffler function. We have also constructed a new integrable mapping with fractional difference.
5. We have found Green and Poisson functions for a biharmonic operator on a disk. Their integral representations are also found.
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