Research for decay and blow-up of solutions to nonlinear Schrodinger equations
| Project/Area Number |
17K05305
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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 | Kumamoto University |
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Principal Investigator |
Kita Naoyasu 熊本大学, 大学院先端科学研究部(工), 教授 (70336056)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 非線型分散型方程式 / 解の減衰評価 / 解の漸近挙動 / 非線型シュレーディンガー方程式 / 解の爆発 / 消散型非線形シュレディンガー方程式 / 解の減衰オーダー / 増幅型非線形シュレディンガー方程式 / 非線形シュレデインガー方程式 / Benjamin-Ono方程式 / 非線形シュレディンガー方程式 / 偏微分方程式 / 漸近挙動 / 爆発解 / 外貨の変動 / エリオット波動の原理 / 解の減衰 |
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| Outline of Final Research Achievements |
In this research, we considered decay rates of the solutions to dissipative nonlinear Schrodinger equations. A nonlinear Schrodinger equation modes the evolution of pulses (or electro-magnetic waves) propagating through optical fibers. In particular, the model treated in this research describes how a pulse is weaken by the impurities lying the optical fiber. We could reveal that, if the power of nonlinearity is Barab-Ozawa’s critical (or sub-critical), the L∞-norm of the solution decays like t{-1/2} (log t){-1/2}, and the L2-norm decays dependently on the regularity of the data but the decay rate tends to (log t){-1/2} as the regularity of the data is refined. In our research, the optimality of these decay rates is also proved. The word “optimality” means that, if a solution decays more rapidly than t{-1/2} (log t){-1/2} in L∞ (or (log t){-1/2} in Ｌ2), then the solution must be trivial (identically equal to 0).
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| Academic Significance and Societal Importance of the Research Achievements |
【産業的な意義】光ファイバーの中を伝わる信号が何km伝わるごとにその強さが半分になるのか，定量的に算出することができた。これは，信号増幅器を何Kmおきに設置すれば良いのか見積もりができるという点で意義がある。数値シミュレーションで信号の減衰を予測する場合には，差分化の精度やプログラムの安定性が懸案になるため，結果の信頼性に疑問が付きまとう。しかし，数学的な解析によって得られた結果には，そのような不備が無いところに利点がある。
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Report
(7 results)
Research Products
(31 results)
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[Book] 微分積分学入門2023
Author(s)
辻川 亨, 北 直泰
Total Pages
270
Publisher
学術図書出版社
ISBN
9784780611243
Related Report
-
[Book] 微分積分学入門2022
Author(s)
辻川亨, 北 直泰
Total Pages
264
Publisher
学術図書出版社
ISBN
9784780609035
Related Report
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[Book] 微分方程式2018
Author(s)
北 直泰
Total Pages
144
Publisher
学術図書出版社
ISBN
9784780606539
Related Report
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