Research of nonlinear dispersive equations with stochastic effects
| Project/Area Number |
15K04944
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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 |
Mathematical analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Research Collaborator |
Adami Riccardo
Holmer Justin
Segawa Etsuo
Anne de Bouard
Poncet Romain
Debussche Arnaud
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
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| Keywords | 解析学 / 非線形分散型方程式 / 散乱 / 時間大域的挙動 / 時間大域挙動 / 解の散乱 / 量子グラフ / 非線形シュレディンガー方程式 / 光通信モデル / 光通信 / 分散性 / 非線形偏微分方程式論 / 確率論 / シュレディンガー方程式 / 量子渦 |
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| Outline of Final Research Achievements |
We studied a model that describes a wave propagation in a one-dimensional linear medium containing a narrow strip of nonlinear material, where the nonlinear strip is assumed to be much smaller than the typical wavelength. This model is used, for example, a wave propagation in nanodevices. We showed that for a large non linearity (called mass super critical), if the initial energy is below the energy of the ground state, the solution scatters in the energy class. Moreover we investigated the asymptotic distribution of the quantum walk associated with this nanodevice model.
On the other hand, we considered the Gross-Pitaevskii equation at positive temperature, where the temperature effect is described by a dissipation and white noise in the equation. We showed that the system converges exponentially to the Gibbs equilibrium as times goes to infinity, in one spatial dimension.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では, ボース・アインシュタイン凝縮や光ファイバー通信のモデルに現れる, 確率的な摂動を伴う非線形シュレディンガー方程式について, 解の挙動, および摂動の特殊解（定在波・渦など) への影響を理論的に解明し, 物理や工学において期待される現象の数学的実証を行うのが目的である. 本研究の成果により, ナノデバイス中にスリットがある場合に内部を伝搬する波がどのような時間大域的挙動をするのか数学により解明された. また, 熱的効果を考慮したボース・アインシュタイン凝縮モデルの凝縮体波動関数の分布が時間無限大で指数的に Gibbs 分布に収束することも厳密に正当化した.
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Report
(5 results)
Research Products
(29 results)
