| Project/Area Number |
26400127
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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 | Kwansei Gakuin University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥130,000 (Direct Cost: ¥100,000、Indirect Cost: ¥30,000)
Fiscal Year 2017: ¥130,000 (Direct Cost: ¥100,000、Indirect Cost: ¥30,000)
Fiscal Year 2016: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000)
Fiscal Year 2015: ¥130,000 (Direct Cost: ¥100,000、Indirect Cost: ¥30,000)
Fiscal Year 2014: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
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| Keywords | 可積分系 / ソリトン / 関数方程式 / 逆散乱法 / リーマン・ヒルベルト問題 / 非線形シュレーディンガー方程式 / 漸近展開 / 離散非線形シュレーディンガー方程式 / 非線形鞍点法 / リーマン･ヒルベルト問題 / 漸近挙動 |
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| Outline of Final Research Achievements |
I studied the focusing integrable discrete nonlinear Schrodinger equation by using the inverse scattering method and the method of nonlinear Schrodinger equation. It has been proved that the soliton resolution conjecture is valid in this case: a reasonable solution splits into a sum of 1-solitons for large time. Phase shift formulas have been obtained. In the timelike region, phase shift is determined by the velocities of solitons and the reflection coefficient. In the spacelike region, it is determined by the velocities of solitons only and the reflection coefficient is irrelevant. Moreover, I studied some Riemann-Hilbert problems under rather general assumptions. It will be a basis of a satisfactory inverse scattering theory of the focusing integrable discrete nonlinear Schrodinger equation.
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| Academic Significance and Societal Importance of the Research Achievements |
離散非線形シュレーディンガー方程式は電子回路のモデルに現れる重要な方程式であるばかりか，離散幾何学（コンピュータグラフィックスとも関係がある）や統計物理など，様々な分野に現れる有名な方程式である．そのような方程式の解の時間無限大における挙動を調べることにより，各種の現象についてより良い理解が得られる．離散非線形シュレーディンガー方程式にもさまざまな種類があるが，なかでも可積分なものについては詳しい情報が得られ，他の場合を調べるときの指針となる．
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