Investigation of symmetries and structure of solutions for nonlinear dispersive equation

• Project/Area Number: 18740074
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Hokkaido University (2008); Kobe University (2006-2007)
• Principal Investigator: TAKAOKA Hideo Hokkaido University, 大学院理学研究院, 教授 (10322794)
• 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: ¥1,200,000 (Direct Cost: ¥1,200,000); Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
• Keywords: 関数方程式 / 関数解析 / 非線形分散型方程式 / 初期値問題 / 適切性

Research Abstract

非線形波動現象を記述する非線形分散型方程式において、解の構造は非線形相互作用によって引き起こされる特異性により特徴付けられる。
このような特異性の考察を、非線形評価式を構成することによって捉え、解の一意存在、データの連続依存性といった初期値問題の適切性の問題、さらに散乱問題などの解の大域挙動の研究を、幾つかの方程式に対して行った。
その結果、初期値問題の適切性と解の大域性に関する技術と理論を進展させた。

Research Products

• [Journal Article] Stability in H1/2 of the sum of K solitons for the Benjamin-Ono equation (2009)
  • Author(s): S. Gustafson, H. Takaoka and T.-P. Tsai
  • Journal Title: J. of Math. Phys 50
  • Peer Reviewed
• [Journal Article] Resonant decompositions and the I-method for the cubic nonlinear Schrodinger equation on R2 Discrete Contin (2008)
  • Author(s): J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
  • Journal Title: Dyn. Syst 21 Pages: 665-686
  • Peer Reviewed
• [Journal Article] Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R^3 (2008)
  • Author(s): J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
  • Journal Title: Ann of Math 167 Pages: 767-865
  • Peer Reviewed
• [Journal Article] Global analysis of nonlinear dispersive-type wave equations (2008)
  • Author(s): H. Takaoka
  • Journal Title: Sugaku 60 Pages: 337-351
  • Peer Reviewed
• [Journal Article] Bilinear Strichartz estimates and applications to the cubic nonlinear Schrodinger equation in two space dimensions (2008)
  • Author(s): H. Takaoka
  • Journal Title: Hokkaido Math. J 37 Pages: 861-870
  • Peer Reviewed
• [Journal Article] Bilinear Strichartz estimates and applications to the cubic nonlinear Schrodinger equation in two space dimensions (2008)
  • Author(s): H. Takaoka
  • Journal Title: Hokkaido Math. J. 37 Pages: 861-870
  • Peer Reviewed
• [Journal Article] Global well-posedness and scattering for the energy-critical nonlinear Schrodinger equation in R^3 (2008)
  • Author(s): H. Takaoka
  • Journal Title: Ann. of Math 167 Pages: 767-865
  • Peer Reviewed
• [Journal Article] Resonant decompositions and the I-method for the cubic nonlinear Schrodinger eauation on R^2 (2008)
  • Author(s): H. Takaoka
  • Journal Title: Discrete Contin. Dyn. Syst 21 Pages: 665-686
  • Peer Reviewed
• [Journal Article] The energy-critical nonlinear Schrodinger equation in R3 Contemp (2007)
  • Author(s): J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
  • Journal Title: Math 439 Pages: 69-80
  • Peer Reviewed
• [Journal Article] The energy-critical nonlinear Schrodinger equation in R^3 (2007)
  • Author(s): 高岡 秀夫
  • Journal Title: Contemp. Math. 439 Pages: 69-80
  • Peer Reviewed
• [Journal Article] Global wellposedness of the modified Benjamin-Ono equation with initial data in H1/2 (2006)
  • Author(s): C. E. Kenig and H. Takaoka
  • Journal Title: Int. Math. Res. Not ID95702 Pages: 44-44
  • Peer Reviewed
• [Journal Article] Global wellposedness of the modified Benjamin-Ono equation with initial data in H^<1/2> (2006)
  • Author(s): H.Takaoka
  • Journal Title: Int. Math. Res. Not. ID 95702 Pages: 44-44
• [Presentation] ベンジャミン・小野方程式に対するソリトン解の安定性について (2008)
  • Author(s): H. Takaoka
  • Organizer: 解析セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 2008-12-11
• [Presentation] Morawetz 評価式と非線形シュレディンガー方程式の散乱理論 (2008)
  • Author(s): H. Takaoka
  • Organizer: 日本数学会北海道支部講演会
  • Place of Presentation: 北海道大学
  • Year and Date: 2008-11-28
• [Presentation] Morawetz評価式と非線形シュレディンガー方程式の散乱理論 (2008)
  • Author(s): 高岡秀夫
  • Organizer: 日本数学会北海道支部講演会
  • Place of Presentation: 北海道大学
  • Year and Date: 2008-11-28
• [Presentation] 非線形分散型波動方程式の大域解析 (2008)
  • Author(s): H. Takaoka
  • Organizer: 日本数学会年会総合講演
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-24
• [Presentation] Bilinear Strichartz estimates and applications to 2D NLS (2007)
  • Author(s): H. Takaoka
  • Organizer: Nonlinear Wave Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2007-08-29
• [Presentation] On the finite dimensional approximation of solution for the KdV equation on the torus (2006)
  • Author(s): H. Takaoka
  • 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
  • Year and Date: 2006-07-05
• [Presentation] 非線形波動方程式の波数間相互作用と解の正則性 (2006)
  • Author(s): H. Takaoka
  • Organizer: 数学教室談話会
  • Place of Presentation: 神戸大学
  • Year and Date: 2006-05-26
• [Presentation] Nonsqueezing result of KdV on the torus (2006)
  • Author(s): H. Takaoka
  • Organizer: 解析セミナー
  • Place of Presentation: 神戸大学
  • Year and Date: 2006-04-20