Global dispersion of nonlinear waves

• Project/Area Number: 18740072
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: NAKANISHI Kenji Kyoto University, 理学研究科, 准教授 (40322200)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,800,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,300,000); Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
• Keywords: 非線形波動 / 分散性 / 漸近解析 / 散乱理論 / 解の一意性 / 漸近安定性 / プラズマ / 超流動 / 水面波 / 時空大域評価 / 非線形エネルギー / 双線形評価 / 特異極限

Research Abstract

様々な波動現象を普遍的に記述する非線形シュレディンガー方程式を中心に、非線形分散型方程式の解の挙動を調べた。特に、プラズマ波動の記述について高階近似モデルからの極限移行を広汎なクラスの解で示し、典型的な特解として反発性の場合に3次元平面波の漸近安定性、また球面値の場合に調和写像について漸近安定性を示した。
また非線形クライン・ゴルドン方程式について散乱・爆発解の分類、非線形シュレディンガー方程式へのノルム移行、調和写像熱流における調和写像の永久振動などの結果も得た。これらの過程で、線形評価と非線形評価を相補的に組み合わせ、挙動の異なる成分間の相互作用や異なる状態への遷移を捉える手法をいくつか考案した。

Research Products

• [Journal Article] Transfer of global wellposedness from nonlinear Klein-Gordon equation to nonlinear Schrodinger equation (2008)
  • Author(s): Kenji Nakanishi
  • Journal Title: Hokkaido Math. J. 37, no.4 Pages: 749-771
  • Peer Reviewed
• [Journal Article] Energy convergence for singular limits of Zakharov type systems (2008)
  • Author(s): Nader Masmoudi and Kenji Nakanishi
  • Journal Title: Invent. Math. 172, no.3 Pages: 535-583
  • Peer Reviewed
• [Journal Article] Large time decay and scattering for wave maps. (2008)
  • Author(s): Joachim Krieger and Kenji Nakanishi
  • Journal Title: Dyn. Partial Differ. Equ. 5, no.1 Pages: 1-37
  • Peer Reviewed
• [Journal Article] Energy convergence for singular limits of Zakharov type systems (2008)
  • Author(s): Nader Masmoudi and Kenji Nakanishi
  • Journal Title: Invent Math 172 Pages: 535-583
  • Peer Reviewed
• [Journal Article] Large time decay and scattering for wave maps (2008)
  • Author(s): Joachim Krieger and Kenji Nakanishi
  • Journal Title: Dyn. Partial Differ. Equ. 5 Pages: 1-37
  • Peer Reviewed
• [Journal Article] Transfer of global wellposedness from nonlinear Klein-Gordon equation to nonlinear Schrodinger equation (2008)
  • Author(s): Kenji Nakanishi
  • Journal Title: Hokkaido Math. J. 37 Pages: 749-771
  • Peer Reviewed
• [Journal Article] Energy scattering for singular limits of Zakharov type systems (2008)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: Inventiones Mathematicae (2008年掲載予定)
  • Peer Reviewed
• [Journal Article] Spectra of linearized operators for NLS solitary waves (2007)
  • Author(s): Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai
  • Journal Title: SIAM J. Math. Anal. 39, no.4 Pages: 1070-1111
  • Peer Reviewed
• [Journal Article] Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. (2007)
  • Author(s): Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai
  • Journal Title: Ann. Henri Poincare 8, no.7 Pages: 1303-1331
  • Peer Reviewed
• [Journal Article] Spectra of linearized operators for NLS solitary waves (2007)
  • Author(s): Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, 〓
  • Journal Title: SIAM Journal of Mathematical Analysis 39 Pages: 1070-1111
  • Peer Reviewed
• [Journal Article] Scattering for the 2D energy critical wave equation
  • Author(s): Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi and Kenji Nakanishi
  • Journal Title: Duke Math. J. (掲載決定)
  • Peer Reviewed
• [Journal Article] Scattering theory for the Gross-Pitaevskii equation in three dimensions
  • Author(s): Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai
  • Journal Title: Commun. Contemp. Math. (掲載決定)
  • Peer Reviewed
• [Journal Article] Uniqueness of solutions for Zakharov systems
  • Author(s): Nader Masmoudi and Kenji Nakanishi
  • Journal Title: Funkcial. Ekvac. (掲載決定)
  • NAID: 130000140521
  • Peer Reviewed
• [Presentation] Scattering and blow-up for nonlinear Klein-Gordon equations (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nagoya Workshop on Differential Equations
  • Place of Presentation: 名古屋大学
  • Year and Date: 2009-02-03
• [Presentation] Asymptotic stability and pulsation of harmonic maps for the Landau-Lifshitz-Gilbert equation (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear Wave and Dispersive Equations
  • Place of Presentation: 京都大学
  • Year and Date: 2009-01-26
• [Presentation] Asymptotic stability of plane waves for the nonlinear Schrodinger equation (2008)
  • Author(s): Kenji Nakanishi
  • Organizer: Linear and Nonlinear Waves, No.6
  • Place of Presentation: ピアザ近江
  • Year and Date: 2008-12-09
• [Presentation] Scattering for the 3D Gross-Pitaevskii equation (2008)
  • Author(s): Kenji Nakanishi
  • Organizer: The Gross-Pitaevskii equation and related topics
  • Place of Presentation: Porquerolles, フランス.
  • Year and Date: 2008-10-16
• [Presentation] Scattering for the 3D Gross-Pitaevskii equation (2008)
  • Author(s): Kenji Nakanishi
  • Organizer: The Gross-Pitaevskii equation and related topics
  • Place of Presentation: Porquerolles
  • Year and Date: 2008-10-16
• [Presentation] Scattering for the Gross-Pitaevskii equation in three dimensions (2008)
  • Author(s): Kenji Nakanishi
  • Organizer: Conference on Partial Differential Equaionts and Applications
  • Place of Presentation: Hammamet, チュニジア
  • Year and Date: 2008-03-28
• [Presentation] Uniqueness of weak solutions for Zakharov-type systems (2008)
  • Author(s): Kenji Nakanishi
  • Organizer: Young Asian Conference on Partial Differential Equations
  • Place of Presentation: Postech, 韓国
  • Year and Date: 2008-01-28
• [Presentation] Asymptotic dynamics for nonlinear Schrodinger equations, I, II (2008)
  • Author(s): 中西賢次
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学
• [Presentation] Scattering for the Gross-Pitaevskii equation (2007)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear Waves and Dispersive Equations
  • Place of Presentation: Oberwolfach, ドイツ
  • Year and Date: 2007-09-09
• [Presentation] Scattering theory for the nonlinear Klein-Gordon equations in the Sobolev critical case in two dimensions (2007)
  • Author(s): Kenji Nakanishi
  • Organizer: Non-linear hyperbolic equations and related topics
  • Place of Presentation: Centro De Giorgi, イタリア
  • Year and Date: 2007-09-07
• [Presentation] Scattering theory for the nonlinear Klein-Gordon equations in the Sobolev critical case in two dimensions (2007)
  • Author(s): 中西賢次
  • Organizer: Non-linear hyperbolic equations and related topics
  • Place of Presentation: Sucuola Normale Superiore (Pisa)
  • Year and Date: 2007-09-07
• [Presentation] 非線形分散波動の漸近解析 (2007)
  • Author(s): 中西賢次
  • Organizer: 日本数学会年会(総合講演)
  • Place of Presentation: 埼玉大学
  • Year and Date: 2007-03-28
• [Presentation] Energy convergence in various singular limits of the Zakharov system (2006)
  • Author(s): 中西賢次
  • Organizer: Linear and Nonlinear Waves, No.4
  • Place of Presentation: 大阪大学
  • Year and Date: 2006-11-06
• [Presentation] Global dispersive solutions for the Gross-Pitaevskii equation (2006)
  • Author(s): Kenji Nakanishi
  • Organizer: AIMS' Sixth International Conference on Dynamical Sytems, Differential Equations and Applications
  • Place of Presentation: Poitiers, フランス
  • Year and Date: 2006-06-25
• [Presentation] Global dispersive solutions for the Gross-Pitaevskii equation in two spatial dimensions (2006)
  • Author(s): Kenji Nakanishi
  • Organizer: The Gross-Pitaevskii and related equations with non-zero boundary conditions at infinity
  • Place of Presentation: Wolfgang Pauli Institute, オーストリア
  • Year and Date: 2006-06-14