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Relations between properties of solutions and geometric symmetry of solutions for nonlinear wave and dispersive equations

Research Project

Project/Area Number 19204012
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

TSUTSUMI Yoshio  京都大学, 大学院・理学研究科, 教授 (10180027)

Co-Investigator(Kenkyū-buntansha) SHIGEKAWA Ichiro  京都大学, 大学院・理学研究科, 教授 (00127234)
KOKUBU Hiroshi  京都大学, 大学院・理学研究科, 教授 (50202057)
NISHIWADA Kimimasa  京都大学, 大学院・理学研究科, 教授 (60093291)
NAKANISHI Kenji  京都大学, 大学院・理学研究科, 准教授 (40322200)
OKAJI Takashi  京都大学, 大学院・理学研究科, 准教授 (20160426)
OHTA Masahito  埼玉大学, 大学院・理学研究科, 准教授 (00291394)
TAKAOKA Hideo  神戸大学, 大学院・理学研究科, 准教授 (10322794)
TSUGAWA Kotaro  名古屋大学, 大学院・多元数理科学研究科, 准教授
Co-Investigator(Renkei-kenkyūsha) OHTA Masahito  埼玉大学, 大学院・理学研究科, 准教授 (00291394)
TAKAOKA Hideo  神戸大学, 大学院・理学研究科, 准教授 (10322794)
TSUGAWA Kotaro  名古屋大学, 大学院・多元数理科学研究科, 准教授
Project Period (FY) 2007 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥41,860,000 (Direct Cost: ¥32,200,000、Indirect Cost: ¥9,660,000)
Fiscal Year 2010: ¥9,620,000 (Direct Cost: ¥7,400,000、Indirect Cost: ¥2,220,000)
Fiscal Year 2009: ¥11,180,000 (Direct Cost: ¥8,600,000、Indirect Cost: ¥2,580,000)
Fiscal Year 2008: ¥9,360,000 (Direct Cost: ¥7,200,000、Indirect Cost: ¥2,160,000)
Fiscal Year 2007: ¥11,700,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥2,700,000)
Keywords非線形波動・分散型方程式 / 初期値問題の適切性・不適切性 / 無条件一意性 / 解の漸近挙動 / フーリエ制限法 / シュレディンガー写像方程式 / 調和写像の熱流 / 調和写像の安定性 / 解の無限振動 / Lugiato-Lefever方程式 / 解の分岐 / 安定定常解 / 中心多様体縮約 / 函数方程式論 / 函数解析学 / 実関数論 / 数理物理学 / 非線形分散型方程式 / 初期値問題の適切性 / 非線形相互作用の構造 / 定常解の分岐 / 空洞ソリトン / 安定性・不安定性 / 修正KdV方程式 / フーリエ制限ノルム / 非線形シュレディンガー方程式 / 超関数の積の評価式 / 斉次ベゾフ空間 / 負の指数のソボレフ空間 / スケール不変性
Research Abstract

The research results are as follows.(1) We proved the stability and instability of vortices for the harmonic map heat-flows, the Landau-Lifshitz equations and the Schrodinger maps.(2) We proved the unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger with powernonlinearity.(3) We obtained the results about the linear stability and the linear instability of stationary solution bifurcating from the constant stationary solution for the Lugiato-Lefever equation, which is the nonlinear Schrodinger equation with damping and forcing.

Report

(6 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report   Self-evaluation Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (32 results)

All 2011 2010 2009 2008 2007

All Journal Article (23 results) (of which Peer Reviewed: 23 results) Presentation (9 results)

  • [Journal Article] Asymptotic stability, concentration, and oscillation in harmonic map Heat-flow, Landau-Lifshitz, and Schrodinger maps on$ R^2$2010

    • Author(s)
      S. Gustafson, K. Nakanishi and T.-P. Tsai
    • Journal Title

      Comm. Math. Phys

      Volume: 300 Pages: 205-242

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Bifurcation analysis to the Lugiato-Lefever equation in one space dimension2010

    • Author(s)
      T. Miyaji, I. Ohnishi and Y. Tsutsumi
    • Journal Title

      Physica D

      Volume: 239 Pages: 2066-2083

    • Related Report
      2010 Annual Research Report 2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition2010

    • Author(s)
      K. Nakanishi, H. Takaoka and Y. Tsutsumi
    • Journal Title

      Discr. Cont. Dynam. Syst

      Volume: 28 Pages: 1635-1654

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic stability, concentration, and oscillation in harmonic map Heat-flow, Landau-Lifshitz, and Schrodinger maps on $R^2$2010

    • Author(s)
      S.Gustafson, K.Nakanishi, T.-P.Tsai
    • Journal Title

      Comm.Math.Phys.

      Volume: 300 Pages: 205-242

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition2010

    • Author(s)
      K. Nakanishi, H. Takaoka, Y. Tsutsumi
    • Journal Title

      Disr. Cont. Dynam. Syst. (掲載確定)

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition2010

    • Author(s)
      K.Nakanishi, H.Takaoka, Y.Tsutsumi
    • Journal Title

      Discr. Cont, Dynam. Syst. (掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A database schema for the analysis of global dynamics of multiparameter systems2009

    • Author(s)
      Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk
    • Journal Title

      SIAM J. Appl. Dyn. Syst

      Volume: 8 Pages: 757-789

    • NAID

      120001629760

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Scattering for the two-dimensional energy-critical wave equation2009

    • Author(s)
      S. Ibrahim, M. Mohamed, N. Masmoude and K. Nakanishi
    • Journal Title

      Duke Math. J.

      Volume: 150 Pages: 287-329

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Scattering for the two-dimensional energy-critical wave equation2009

    • Author(s)
      S. Ibrahim, M. Mohamed, N. Masmoudi, K. Nakanishi
    • Journal Title

      Duke Math. J. 150

      Pages: 287-329

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] A database schema for the analysis of global dynamics of multiparameter systems, SIAM2009

    • Author(s)
      Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk
    • Journal Title

      J. Appl. Dyn. Syst. 8

      Pages: 757-789

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Scattering for the two-dimensional energy-critical wave equation2009

    • Author(s)
      S.Ibrahim, M.Mohamed, N.Masmoudi, K.Nakanishi
    • Journal Title

      Duke Math. J. 150

      Pages: 287-329

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A database schema for the analysis of global dynamics of multiparameter systems2009

    • Author(s)
      Z.Arai, W.Kalies, H.Kokubu, K.Mischaikow, H.Oka, P.Pilarczyk
    • Journal Title

      SIAM J.Appl. Dyn, Syst. 8

      Pages: 757-789

    • NAID

      120001629760

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation2008

    • Author(s)
      Y. Y. Su Win and Y. Tsutsumi
    • Journal Title

      Hokkaido Math. J.

      Volume: 37 Pages: 839-859

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Time decay of solution for the KdV equation with multiplicative space-time noise2008

    • Author(s)
      Y. Tsutsumi
    • Journal Title

      Differential and Integral Equations

      Volume: 21 Pages: 959-970

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation2008

    • Author(s)
      Yin Yin Su Win, Y. Tsutsumi
    • Journal Title

      Hokkaido Math. J. 37

      Pages: 839-859

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Time decay of solution for the KdV equation with multiplicative space-time noise2008

    • Author(s)
      Y. Tsutsumi
    • Journal Title

      Differential and Integral Equations. 21

      Pages: 959-970

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation2008

    • Author(s)
      Y. Y. Su Win, Y. Tsutsumi
    • Journal Title

      Hokkaido Mathematical Journal 37

      Pages: 839-859

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global dispersive solutions of the Gross-Pitaevskii equation in two and three dimensions2007

    • Author(s)
      S. Gustafson, K. Nakanishi and T.-P. Tsai
    • Journal Title

      Ann. Henri Poincare

      Volume: 8 Pages: 1303-1331

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Rigorous verification of cocoon bifurcations in the Michelson system2007

    • Author(s)
      H. Kokubu, D. Wilczak and P. Zigliczynski
    • Journal Title

      Nonlinearity

      Volume: 20 Pages: 2147-2174

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Stability of solitary waves for the Ostrovsky equation2007

    • Author(s)
      Y. Liu and M. Ohta
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 136 Pages: 511-517

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Stability of solitary waves for the Ostrovsky equation2007

    • Author(s)
      Y. Liu and M. Ohta
    • Journal Title

      Proc. Amer. Math. Soc. 136

      Pages: 511-517

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Rigorous verification of cocoon bifurcations in the Michelson system2007

    • Author(s)
      H. Kokubu, D. Wilczak and P. Zigliczynski
    • Journal Title

      Nonlinearity 20

      Pages: 2147-2174

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions2007

    • Author(s)
      S. Gustafson, K. Nakanishi and T.-P. Tsai
    • Journal Title

      Ann. Henri Poincare 8

      Pages: 1303-1331

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] 1D quantic NLS with white noise2011

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Nonlinear Phenomena : A View from Mathematics and Physics
    • Place of Presentation
      国立台湾大学(台湾)
    • Year and Date
      2011-01-14
    • Related Report
      2010 Final Research Report
  • [Presentation] 1D quantic NLS with white noise2011

    • Author(s)
      堤誉志雄
    • Organizer
      Nonlinear Phenomena : A View from Mathematics and Physics
    • Place of Presentation
      国立台湾大学(台湾)(招待講演)
    • Year and Date
      2011-01-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] Stability of cavity soliton for the Lugiato-Lefever equation with additive noise2009

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Analysis of Nonlinear Wave Equations and Applications in Engineerign
    • Place of Presentation
      BIRS, Banff, Canada.
    • Year and Date
      2009-08-10
    • Related Report
      2010 Final Research Report
  • [Presentation] Stability of cavity soliton for the Lugiato-Lefever equation with additive noise2009

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Analysis of Nonlinear Wave Equations and Applications in Engineering
    • Place of Presentation
      BIRS, Banff, Canada
    • Year and Date
      2009-08-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrodinger equation2008

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Asymptotics and Singularities in Nonlinear and Geometric Dispersive Equations
    • Place of Presentation
      BIRS, Banff, Canada.
    • Year and Date
      2008-08-28
    • Related Report
      2010 Final Research Report
  • [Presentation] Unconditional uniqueness of solution for the Cauohy problem of the nonlinear Schrodinger equation2008

    • Author(s)
      Y. Tsutsunii
    • Organizer
      Asymptotics and Singularities in Nonlinear and Geometric Dispersive Equations
    • Place of Presentation
      Banff, Canada
    • Year and Date
      2008-08-28
    • Related Report
      2008 Annual Research Report
  • [Presentation] Unconditional uniqueness of solution for the Cauchy problem of nonlinear Schrodinger equation2007

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Nonlinear Hyperbolic Equations and Related Topics
    • Place of Presentation
      University of Pisa, Pisa, Italy.
    • Year and Date
      2007-09-06
    • Related Report
      2010 Final Research Report
  • [Presentation] Unconditional uniqueness of solution for the Cauchy problem of nonlinear Schrodinger equation2007

    • Author(s)
      Y. Tsutsumi
    • Organizer
      Nonlinear Hyperbolic Equations and Related Topics
    • Place of Presentation
      ピザ大学 (イタリア)
    • Year and Date
      2007-09-06
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Unconditional uniqueness of solution for the Cauchy problem of nonlinear Schrodinger equation2007

    • Author(s)
      堤誉志雄
    • Organizer
      Nonlinear Hyperbolic Equations and Related Topics
    • Place of Presentation
      ピザ大学(イタリア)
    • Year and Date
      2007-09-06
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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