Study of long time behavior of nonlinear dispersive equations via Hamiltonian structure

• Project/Area Number: 15K17568
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Chiba University
• Principal Investigator: Masaya Maeda 千葉大学, 大学院理学研究院, 准教授 (40615001)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); 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); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 非線形分散型方程式 / ソリトン / 漸近安定性 / 非線形シュレディンガー方程式 / 量子ウォーク / 国際情報交換 / 離散シュレディンガー方程式 / 散乱 / 離散非線形シュレディンガー方程式 / 定在波解 / 励起状態解 / 軌道安定性 / フェルミ黄金律

Outline of Final Research Achievements

I have studied the long time behavior of solutions of nonlinear dispersive equations such as nonlinear Schrodinger equations. In particular, I studied soliton solutions and proved the soliton decomposition for small solutions and revealed the instability mechanism induced by nonlinear Fermi Golden rule for vortex solutions.

Academic Significance and Societal Importance of the Research Achievements

ソリトン分解予想は非線形分散型方程式論における重要な予想であるがその解決は未だ程遠い。本研究課題ではいくつかの特別な状況における非線形分散型方程式の解の大域挙動を解明することによりソリトン分解予想に対して一定の貢献を行った。また、光ファイバー等の研究に登場するボルテックス解の不安定性の解明は工学的にも重要であると思われる。

Research Products

• [Int'l Joint Research] トリエステ大学(イタリア)
• [Int'l Joint Research] トリエステ大学(イタリア)
• [Int'l Joint Research] トリエステ大学(イタリア)
• [Int'l Joint Research] テネシー大学(米国)
• [Int'l Joint Research] トリエステ大学(イタリア)
• [Int'l Joint Research] テネシー大学(米国)
• [Journal Article] Scattering and inverse scattering for nonlinear quantum walks (2018)
  • Author(s): Maeda Masaya, Sasaki Hironobu, Segawa Etsuo、Suzuki Akito、Suzuki Kanako
  • Journal Title: Discrete & Continuous Dynamical Systems - A Volume: 38 Issue: 7 Pages: 3687-3703
  • DOI: 10.3934/dcds.2018159
  • Peer Reviewed
• [Journal Article] Weak limit theorem for a nonlinear quantum walk (2018)
  • Author(s): M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki
  • Journal Title: Quantum Inf Process Volume: 17 Issue: 9 Pages: 215-215
  • DOI: 10.1007/s11128-018-1981-z
  • Peer Reviewed
• [Journal Article] On Nonlinear Profile Decompositions and Scattering for an NLS-ODE Model (2018)
  • Author(s): Cuccagna Scipio、Maeda Masaya
  • Journal Title: International Mathematics Research Notices Volume: 印刷中 Issue: 18 Pages: 5679-5722
  • DOI: 10.1093/imrn/rny173
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Existence and asymptotic stability of quasi-periodic solution of discrete NLS with potential in Z (2017)
  • Author(s): Masaya Maeda
  • Journal Title: SIAM J. Math. Anal. Volume: 49 Issue: 5 Pages: 3396-3426
  • DOI: 10.1137/16m1069729
  • Peer Reviewed
• [Journal Article] On Orbital Instability of Spectrally Stable Vortices of the NLS in the Plane (2016)
  • Author(s): Scipio Cuccagna and Masaya Maeda
  • Journal Title: Journal of Nonlinear Science Volume: 26 Issue: 6 Pages: 1851-1894
  • DOI: 10.1007/s00332-016-9322-9
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On small energy stabilization in the NLKG with a trapping potential (2016)
  • Author(s): Scipio Cuccagna, Masaya Maeda and Tuoc V. Phan
  • Journal Title: Nonlinear Analysis: Theory, Methods & Applications Volume: 146 Pages: 32-58
  • DOI: 10.1016/j.na.2016.08.009
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On small energy stabilization in the NLS with a trapping potential (2015)
  • Author(s): Scipio Cuccagna and Masaya Maeda
  • Journal Title: Analysis and PDE Volume: 8 Issue: 6 Pages: 1289-1349
  • DOI: 10.2140/apde.2015.8.1289
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On weak interaction between a ground state and a trapping potential (2015)
  • Author(s): Scipio Cuccagna and Masaya Maeda
  • Journal Title: Discrete and Continuous Dynamical Systems - Series A Volume: 35 Issue: 8 Pages: 3343-3376
  • DOI: 10.3934/dcds.2015.35.3343
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Oscillation of soliton around critical frequency (2018)
  • Author(s): 前田昌也
  • Organizer: 京都大学数理解析研究所共同研究 「非線形波動及び分散型方程式の研究」
• [Presentation] On Scattering for NLS-ODE model having metastable solution (2018)
  • Author(s): 前田昌也
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Int'l Joint Research
• [Presentation] Long time oscillation of solutions of nonlinear Schrodinger equations near minimal mass ground state (2018)
  • Author(s): 前田昌也
  • Organizer: 第5回量子渦と非線形波動
  • Invited
• [Presentation] On long time dynamics of small solutions of nonlinear dispersive equations with potential (2017)
  • Author(s): Masaya Maeda
  • Organizer: 2017 Participating school in analysis of PDE Stability of solitons and periodic waves
  • Int'l Joint Research / Invited
• [Presentation] 量子ウォークの分散評価 (2017)
  • Author(s): 前田昌也
  • Organizer: RIMS 関数不等式の最良定数とその周辺
  • Invited
• [Presentation] On small solutions of nonlinear Schrodinger equations with potential (2016)
  • Author(s): 前田昌也
  • Organizer: Workshop on Harmonic Analysis and PDE
  • Place of Presentation: 華中師範大学（中華人民共和国武漢市）
  • Year and Date: 2016-01-10
  • Int'l Joint Research
• [Presentation] On Long Time Dynamics of Small Solutions of Discrete Nonlinear Schrodinger Equations (2016)
  • Author(s): Maeda Masaya
  • Organizer: SIAM Conference on Nonlinear Waves and Coherent Structures
  • Place of Presentation: Sheraton Philadelphia Society Hill One Dock Street, (2nd and Walnut Streets) Philadelphia, Pennsylvania 19106 United States
  • Int'l Joint Research / Invited
• [Presentation] On orbital instability of excited states of nonlinear Schrodinger equations (2016)
  • Author(s): 前田昌也
  • Organizer: 2016 日本数学会 秋季総合分科会
  • Place of Presentation: 関西大学千里山キャンパス（大阪府吹田市）
  • Invited
• [Presentation] On small solutions of continuous and discrete nonlinear Schr¨odinger equation (2015)
  • Author(s): 前田昌也
  • Organizer: The 8th international conference on Science and Mathematics Education in Developing Countries
  • Place of Presentation: ヤンゴン大学（ミャンマー連邦共和国ヤンゴン市）
  • Year and Date: 2015-12-04
  • Int'l Joint Research
• [Presentation] On long time behavior of small data solutions of NLS with potential (2015)
  • Author(s): 前田昌也
  • Organizer: 調和解析と非線形偏微分方程式
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2015-07-08