2005 Fiscal Year Final Research Report Summary
Symmetry and Complete Integrability of Nonlinear Difference Equations
| Project/Area Number |
14340053
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Aoyama Gakuin University (2005)
The University of Tokyo (2002-2004) |
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Principal Investigator |
SATSUMA Junkichi Aoyama Gakuin University, College of Science and Engineering, Professor, 理工学部, 教授 (70093242)
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| Co-Investigator(Kenkyū-buntansha) |
YANO Koichi Aoyama Gakuin University, College of Science and Engineering, Professor., 理工学部, 教授 (60114691)
TOKIHIRO Tetsuji The University of Tokyo, Graduate School of Mathematical Sciences., Professor, 大学院数理科学研究科, 教授 (10163966)
OKAMOTO Kazuo The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理科学研究科, 教授 (40011720)
OHTA Yasuhiro Kobe University, Graduate School of Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (10213745)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Soliton / Complete Integrability / Difference Equations / Painleve Equations / Symmetry / Ultradiscrete System |
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| Research Abstract |
The purpose of this research is to clarify the mathematical structure of singularity confinement condition for nonlinear difference equations, to extend the condition to ultradiscrete systems, and to study algebraic and geometrical structure of discrete integrable systems. Some of our results are as follows.
1. We have studied the relation between continuous and discrete Painleve equations and given a determinant-type solution for them. We also presented a method to describe q-Painleve equations in a unified way. Furthermore, we studied their transformation structure in detail.
2. We have presented a few of new integrable nonlinear equations and studied solutions of a coupled KP equation and the higher order nonlinear Schrodinger equations.
3.We have proposed an ultradiscrete Sine-Gordon equation, which is considered to be a cellular automaton with soliton-type solutions. We have also given an ultradiscrete modified KdV equation and have shown that the equation directly relates with a box and ball system. Moreover, we proposed a method to ultradiscretize variables without positivity.
4.We have studied periodic ultradiscrete systems and determined fundamental period of their solutions exactly and have given an algolithm to constitute the conserved quantities explicitly. We also clarify the relation between conserved quantities of ultradiscrete and continuous systems.
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