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Global dynamics of nonlinear dispersive equations

Research Project

Project/Area Number 25400159
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionOsaka University (2015-2016)
Kyoto University (2013-2014)

Principal Investigator

Nakanishi Kenji  大阪大学, 情報科学研究科, 教授 (40322200)

Research Collaborator Schlag Wilhelm  
Krieger Joachim  
Gustafson Stephen  
Tsai Tai-Peng  
Masmoudi Nader  
Ibrahim Slim  
Guo Zihua  
Project Period (FY) 2013-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords非線形分散型方程式 / 解の大域挙動 / ソリトン / 基底状態 / 励起状態 / 解の爆発 / 非線形散乱理論 / 漸近安定性 / 散乱理論 / 中心安定多様体 / 臨界指数 / 国際研究者交流:フランス / 非線形波動方程式 / 大域ダイナミクス / 爆発解 / 水面波
Outline of Final Research Achievements

Concerning the global dynamics of the nonlinear Schrodinger equation, classification is given in the energy region containing two families of solitons which are very different from each other. More precisely, classification on global behavior of solutions and description of the transition are given under an energy constraint such that the ground state solitons and the first excited state solitons are contained, which are generated mainly from the interaction with the linear and the nonlinear potentials, respectively. Moreover, partial results are obtained on classification of global behavior of solutions for the generalized KdV equations, the water wave equations, the Zakharov systems and the Gross-Pitaevskii equations.

Report

(5 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • 2013 Research-status Report
  • Research Products

    (33 results)

All 2016 2015 2014 2013 Other

All Int'l Joint Research (5 results) Journal Article (12 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 12 results) Presentation (15 results) (of which Int'l Joint Research: 7 results,  Invited: 14 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Monash University(オーストラリア)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] Georgia Institute of Technology(米国)

    • Related Report
      2016 Annual Research Report
  • [Int'l Joint Research] UC San Diego(米国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Universitat Bielefeld(ドイツ)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] Monash University(オーストラリア)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Global dynamics above the ground state for the energy-critical Schrodinger equation with radial data2016

    • Author(s)
      Kenji Nakanishi and Tristan Roy
    • Journal Title

      Commun. Pure Appl. Anal.

      Volume: 15 Issue: 6 Pages: 2023-2058

    • DOI

      10.3934/cpaa.2016026

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Codimension One Threshold Manifold for the Critical gKdV Equation2016

    • Author(s)
      Yvan Martel , Frank Merle, Kenji Nakanishi, Pierre Raphael (最後のeはトレマが付く)
    • Journal Title

      Communications in Mathematical Physics

      Volume: 342 Issue: 3 Pages: 1075-1106

    • DOI

      10.1007/s00220-015-2509-3

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics2015

    • Author(s)
      Chongsheng Cao, Slim Ibrahim, Kenji Nakanishi, Edriss S. Titi
    • Journal Title

      Communications in Mathematical Physics

      Volume: 337 Issue: 2 Pages: 473-482

    • DOI

      10.1007/s00220-015-2365-1

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Well-posedness and scattering for the Zakharov system in four dimensions2015

    • Author(s)
      Ioan Bejenaru, Zihua Guo, Sebastian Herr and Kenji Nakanishi
    • Journal Title

      Analysis & PDE

      Volume: 8 Issue: 8 Pages: 2029-2055

    • DOI

      10.2140/apde.2015.8.2029

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sharp Global Regularity for the 2 + 1-Dimensional Equivariant Faddeev Model2015

    • Author(s)
      Dan-Andrei Geba, Kenji Nakanishi and Xiang Zhang
    • Journal Title

      International Mathematical Research Notices

      Volume: 2015 Pages: 1-17

    • DOI

      10.1093/imrn/rnv037

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Trudinger–Moser inequality on the whole plane with the exact growth condition2015

    • Author(s)
      Slim Ibrahim, Nader Masmoudi and Kenji Nakanishi
    • Journal Title

      Journal of the European Mathematical Society

      Volume: 17 Issue: 4 Pages: 819-835

    • DOI

      10.4171/jems/519

    • NAID

      120005666188

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Center-stable manifold of the ground state in the energy space for the critical wave equation2015

    • Author(s)
      Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
    • Journal Title

      Mathematische Annalen

      Volume: 361 Issue: 1-2 Pages: 1-50

    • DOI

      10.1007/s00208-014-1059-x

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation2014

    • Author(s)
      Slim Ibrahim, Nader Masmoudi and Kenji Nakanishi
    • Journal Title

      Transactions of the AMS

      Volume: 366 Pages: 5653-5669

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions2014

    • Author(s)
      Joachim Krieger, Wilhelm Schlag and Kenji Nakanishi
    • Journal Title

      Comm. Math. Phys.

      Volume: 327 Issue: 1 Pages: 309-332

    • DOI

      10.1007/s00220-014-1900-9

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Global dynamics below the ground state energy for the Klein-Gordon-Zakharov system in the 3D radial case2014

    • Author(s)
      Zihua Guo, Kenji Nakanishi and Shuxia Wang
    • Journal Title

      Communications in Partial Differential Equations

      Volume: 39 Issue: 6 Pages: 1158-1184

    • DOI

      10.1080/03605302.2013.836715

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Generalized Strichartz estimates and scattering for 3D Zakharov system2014

    • Author(s)
      Z.-H. Guo, S.-H. Lee, K. Nakanishi and C.-B. Wang
    • Journal Title

      Comm. Math. Phys.

      Volume: 331 Issue: 1 Pages: 239-259

    • DOI

      10.1007/s00220-014-2006-0

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multifrequency NLS scaling for a model equation of gravity-capillary waves2013

    • Author(s)
      Nader Masmoudi and Kenji Nakanishi
    • Journal Title

      Comm. Pure Appl. Math.

      Volume: 66 Issue: 8 Pages: 1202-1240

    • DOI

      10.1002/cpa.21464

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Presentation] Global dynamics of the nonlinear Schrodinger equation with a potential2016

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Singularity formation and long-time behavior in dispersive PDEs
    • Place of Presentation
      University of Bonn (ドイツ・ボン)
    • Year and Date
      2016-03-15
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering for the Gross-Pitaevskii equation in the 3D radial energy space2016

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Second Workshop on Evolution Equations in Valdivia
    • Place of Presentation
      Universidad Austral de Chile (チリ・Valdivia
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Maximizers of a Trudinger-Moser-type inequality with the critical growth on the whole plane2016

    • Author(s)
      Kenji Nakanishi
    • Organizer
      偏微分方程式の 解の形状解析
    • Place of Presentation
      数理解析研究所(京都)
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] Wellposedness and scattering for the Zakharov system in four dimensions2016

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Nonlinear Waves 2016
    • Place of Presentation
      Institut des Hautes Etudes Scientifiques (フランス・Bures-sur-Yvette)
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Global dynamics beyond the first excited state of the nonlinear Schrodinger equation with a potential2015

    • Author(s)
      Kenji Nakanishi
    • Organizer
      2015 Annual Meeting of the Taiwan Mathematical Society
    • Place of Presentation
      National University of Kaohsiung (台湾・高雄)
    • Year and Date
      2015-12-20
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering for the Gross-Pitaevskii equation in the energy space2015

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Fundamental Problems in Mathematical and Theoretical Physics
    • Place of Presentation
      早稲田大学 (東京都新宿区)
    • Year and Date
      2015-10-03
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global dynamics of nonlinear dispersive equations2015

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Introductory Workshop: Randomness and long time dynamics in nonlinear evolution differential equations
    • Place of Presentation
      Mathematical Sciences Research Institute (アメリカ・バークレイ)
    • Year and Date
      2015-08-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Scattering and wellposedness for the Zakharov system in 4D2015

    • Author(s)
      Kenji Nakanishi
    • Organizer
      International Conference of Harmonic Analysis and Applications
    • Place of Presentation
      Central China Normal University (中国・武漢)
    • Year and Date
      2015-05-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Classification of global dynamics for nonlinear dispersive equations2015

    • Author(s)
      Kenji Nakanishi
    • Organizer
      JAMI seminar
    • Place of Presentation
      Johns Hopkins University
    • Year and Date
      2015-02-06 – 2015-02-20
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Global dynamics of nonlinear dispersive equations above the ground state energy2014

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Differential Geometry-Mathematical Physics-PDE seminar
    • Place of Presentation
      University of British Columbia
    • Year and Date
      2014-11-04
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Multi-frequency NLS limit for a model equation of water wave2014

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Harmonic Analysis and Partial Differential Equations, closing workshop
    • Place of Presentation
      University of Bonn
    • Year and Date
      2014-08-04 – 2014-08-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Multi-frequency NLS approximation for a water wave model equation2014

    • Author(s)
      中西賢次
    • Organizer
      第31回九州における偏微分方程式研究集会
    • Place of Presentation
      福岡大学メディカルホール
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Multi-frequency NLS approximation for a water wave model equation2014

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Kyoto University-Chung-Ang University Symposium On Nonlinear Partial Differential equations
    • Place of Presentation
      高山市民文化会館
    • Related Report
      2013 Research-status Report
  • [Presentation] Threshold manifold of global dynamics for the critical wave equation2013

    • Author(s)
      Kenji Nakanishi
    • Organizer
      Nonlinear Waves and Dispersive Equations
    • Place of Presentation
      Mathematisches Forschungsinstitut Oberwolfach
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Threshold manifold for global dynamics of the critical wave equation

    • Author(s)
      Kenji Nakanishi
    • Organizer
      SEMINAIRE GEOMETRIE, EDP ET PHYSIQUE MATHEMATIQUE
    • Place of Presentation
      Universite de Cergy-Pontois
    • Related Report
      2013 Research-status Report
    • Invited
  • [Funded Workshop] Nonlinear Wave and Dispersive Equations, Kyoto 20162016

    • Place of Presentation
      京都大学
    • Year and Date
      2016-09-06
    • Related Report
      2016 Annual Research Report

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Published: 2014-07-25   Modified: 2022-02-16  

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