2018 Fiscal Year Final Research Report
An approach to soliton theory by the method of the Sato Grassmannian and multi-variate sigma functions
| Project/Area Number |
15K04907
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tsuda University |
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Principal Investigator |
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| Research Collaborator |
Yamada Yasuhiko
Inoue Rei
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 一般化ソリトン / KP方程式 / 通常3重点 / 佐藤グラスマン / リーマンテータ関数 / 多変数シグマ関数 / タウ関数 / 準周期解 |
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| Outline of Final Research Achievements |
The KP equation describes shallow water waves and is known to possess soliton solutions. It has several other solutions. The class of solutions expressed by theta functions is one of them. They describe periodic solutions for example.A theta function solution is associated with a complex algebraic curve which, geometrically, is considered as a surface with several holes. If we tend the period to infinity of a periodic solution, then the solution approaches to a soliton solution. The KP equation has hidden symmetries. Then a theta function solution has several independent periods. Therefore there are several ways to tend periods to infinity. Consequently various curious solutions can appear as limits of a theta function solution. However there were technical difficulties in taking these limits.
In the present research we have overcome these difficulties by the method of the Sato Grassmannian and computed limits of theta function solutions for various families of plane algebraic curves.
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| Free Research Field |
可積分系とテータ関数
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| Academic Significance and Societal Importance of the Research Achievements |
ソリトンは非線形波動の一種で厳密な解析が可能であることから数学的にも応用上も多方面から研究されてきた。ソリトンを記述する方程式はソリトンの他にも様々な解を持つ。テータ関数解はその一つでソリトンはその極限であると考えられている。極限を厳密に計算することは種々の観点から興味のある問題であるが技術的に難しい問題であった。本研究では佐藤グラスマンの方法という新しい方法を提案しそれが確かにテータ関数解の極限の計算に有効であることを様々な例で実証した。この方法は平坦でない波の上に出来るソリトンやrogue wavesなど最近活発に研究されている問題に新しい研究方法を提供すると期待される。
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