Global analysis of dispersion and resonance of nonlinear waves

• Project/Area Number: 21740095
• 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 京都大学, 理学研究科, 准教授 (40322200)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 関数方程式 / 関数解析 / 実解析 / 非線形波動 / 波の分散性 / 非線形共鳴 / 解の大域挙動 / 解の爆発 / ソリトン / 中心安定多様体 / 分散性 / 漸近解析 / 特異摂動 / 散乱理論 / 大域動力学 / 散乱 / 爆発 / 基底状態 / ワンパス定理 / 初期値問題の適切性 / Trudmger-Moser不等式 / エネルギーの日数減衰

Research Abstract

I constructed a theory to determine space-time behavior of general solutions for partial differential equations describing nonlinear waves, from the initial data and topological information on the solution in the phase space, and completely classified space-time dynamics up to an energy level slightly above that of the ground state. It is a mathematically rigorous description, based solely on the equation, of the mechanism of scattering, soliton and blow-up, which are typical behavior of nonlinear waves, and of transitions among them.

Research Products

• [Journal Article] Global dynamics of thenonradial energy-critical waveequation above the ground state energy (2013)
  • Author(s): Joachim Krieger, Kenji Nakanishi,Wilhelm Schlag
  • Journal Title: Discrete Contin. Dyn. Syst Volume: 33 Issue: 6 Pages: 2423-2450
  • DOI: 10.3934/dcds.2013.33.2423
  • Peer Reviewed
• [Journal Article] Global dynamics below theground state energy for the Zakharovsystem in the 3D radial case (2013)
  • Author(s): Zihua Guo, Kenji Nakanishi, ShuxiaWang
  • Journal Title: Adv.Math Volume: 238 Pages: 412-441
  • DOI: 10.1016/j.aim.2013.02.008
  • Peer Reviewed
• [Journal Article] Small energy scattering for the Zakharov system with radial symmetry (2013)
  • Author(s): Zihua Guo, Kenji Nakanishi
  • Journal Title: Int Math Res Notices Volume: 未定 Issue: 9 Pages: 2327-2342
  • DOI: 10.1093/imrn/rns296
  • NAID: 120005651928
  • Peer Reviewed
• [Journal Article] Global dynamics abovethe ground state energy for theone-dimensional NLKG equation (2012)
  • Author(s): Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag
  • Journal Title: Math.Z. Volume: 272 Issue: 1-2 Pages: 297-316
  • DOI: 10.1007/s00209-011-0934-3
  • Peer Reviewed
• [Journal Article] Invariant Manifolds Around SolitonManifolds for the NonlinearKlein-Gordon Equation (2012)
  • Author(s): Kenji Nakanishi, Wilhelm Schlag
  • Journal Title: SIAM J. Math.Anal Volume: 44 Issue: 2 Pages: 1175-1210
  • DOI: 10.1137/11082720x
  • NAID: 120004288063
  • Peer Reviewed
• [Journal Article] Globaldynamics above the ground stateenergy for the cubic NLS equation in3D (2012)
  • Author(s): K.Nakanishi, W.Schlag
  • Journal Title: Calculus of variations andPartial differential equations Volume: 44 Issue: 1-2 Pages: 1-45
  • DOI: 10.1007/s00526-011-0424-9
  • Peer Reviewed
• [Journal Article] Global Dynamics Above the Ground State for the Nonlinear Klein-Gordon Equation Without a Radial Assumption (2012)
  • Author(s): K.Nakanishi, W.Schlag
  • Journal Title: Archivefor rational mechanics and analysis Volume: 203 Issue: 3 Pages: 809-851
  • DOI: 10.1007/s00205-011-0462-7
  • Peer Reviewed
• [Journal Article] Scattering threshold forthe focusing nonlinear Klein-Gordonequation (2011)
  • Author(s): Slim Ibrahim, Nader Masmoudi, KenjiNakanishi
  • Journal Title: Analysis & PDE Volume: 4 Issue: 3 Pages: 405-460
  • DOI: 10.2140/apde.2011.4.405
  • Peer Reviewed
• [Journal Article] Well-posedness fornonlinear Dirac equations in onedimension (2010)
  • Author(s): Shuji Machihara, Kenji Nakanishi,Kotaro Tsugawa
  • Journal Title: Kyoto Journal ofMathematics Volume: 50 Pages: 403-451
  • Peer Reviewed
• [Journal Article] From the Klein-Gordon-Zakharovsystem to a singular nonlinear (2010)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: Schrodinger system Volume: 27 Pages: 1073-1096
  • Peer Reviewed
• [Journal Article] Twoasymptotic problems for a singularnonlinear Schrodinger system (2010)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: American Journal of Mathematics Volume: 132 Pages: 1311-1336
  • Peer Reviewed
• [Journal Article] Asymptotic stability,concentration, and oscillation inharmonic map heat-flow,Landau-Lifshitz, and Schrodinger maps on R^2 (2010)
  • Author(s): Stephen Gustafson, Kenji Nakanishi,Tai-Peng Tsai
  • Journal Title: Communications inMathematical Physics Volume: 300 Pages: 205-242
  • Peer Reviewed
• [Journal Article] Local well-posedness in lowregularity of the mKdV equation withperiodic boundary condition (2010)
  • Author(s): Kenji Nakanishi, Hideo Takaoka, YoshioTsutsumi
  • Journal Title: Discrete and continuous dynamicalsystems Volume: 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Global dynamics above the ground stateenergy for the focusing nonlinearKlein-Gordon equation (2010)
  • Author(s): Kenji Nakanishi, Wilhelm Schlag
  • Journal Title: Journal ofDifferential Equations Volume: 250 Pages: 2299-2333
  • NAID: 120002770950
  • Peer Reviewed
• [Journal Article] Exponential energy decay fordamped Klein-Gordon equation withnonlinearities of arbitrary growth (2010)
  • Author(s): Lassad Aloui, Slim Ibrahim, KenjiNakanishi
  • Journal Title: Communications in Partial DifferentialEquations Volume: 36 Pages: 797-818
  • Peer Reviewed
• [Journal Article] Well-posedness for nonlinear Dirac equations in one dimension (2010)
  • Author(s): Shuji Machihara, Kenji Nakanishi, Kotaro Tsugawa
  • Journal Title: Kyoto Journal of Mathematics Volume: 50 Pages: 403-451
  • Peer Reviewed
• [Journal Article] From the Klein-Gordon-Zakharov system to a singular nonlinear Schrodinger system (2010)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: Annales de l'Institut Henri Poincare Non Linear Analysis Volume: 27 Pages: 1073-1096
  • Peer Reviewed
• [Journal Article] Two asymptotic problems for a singular nonlinear Schrodinger system (2010)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: American Journal of Mathematics Volume: 132 Pages: 1311-1336
  • Peer Reviewed
• [Journal Article] Asymptotic stability, concentration, and oscillation in harmonic mapheat-flow, Landau-Lifshitz, and Schrodinger maps on R ^ 2 (2010)
  • Author(s): Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai
  • Journal Title: Communications in Mathematical Physics Volume: 300 Pages: 205-242
  • Peer Reviewed
• [Journal Article] Local well-posedness in low regularity of the mKdV equation with periodic boundary condition (2010)
  • Author(s): Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi
  • Journal Title: Discrete and continuous dynamical systems Volume: 28 Pages: 1635-1654
  • Peer Reviewed
• [Journal Article] Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation (2010)
  • Author(s): Kenji Nakanishi, Wilhelm Schlag
  • Journal Title: Journal of Differential Equations Volume: 250 Pages: 2299-2333
  • NAID: 120002770950
  • Peer Reviewed
• [Journal Article] Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth (2010)
  • Author(s): Lassad Aloui, Slim Ibrahim, Kenji Nakanishi
  • Journal Title: Communications in Partial Differential Equations Volume: 36 Pages: 797-818
  • Peer Reviewed
• [Journal Article] Uniqueness of Solutions for ZakharovSystem (2009)
  • Author(s): Nader Masmoudi and Kenji Nakanishi
  • Journal Title: Funkcialaj Ekvacioj Volume: 52 Pages: 233-253
  • NAID: 130000140521
  • Peer Reviewed
• [Journal Article] Scattering for the two-dimensionalenergy-critical wave equation (2009)
  • Author(s): Slim Ibrahim, Mohamed Majdoub, NaderMasmoudi, and Kenji Nakanishi
  • Journal Title: DukeMath. J Volume: 150 Pages: 287-329
  • Peer Reviewed
• [Journal Article] Scattering theory forthe Gross-Pitaevskii equation in threedimensions (2009)
  • Author(s): Stephen Gustafson, Kenji Nakanishi andTai-Peng Tsai
  • Journal Title: Commun. Contemp. Math Volume: 11 Pages: 657-707
  • Peer Reviewed
• [Journal Article] Uniqueness of Solutions for Zakharov System (2009)
  • Author(s): Nader Masmoudi, Kenji Nakanishi
  • Journal Title: Funkcialaj Ekvacioj 52 Pages: 233-253
  • NAID: 130000140521
  • Peer Reviewed
• [Journal Article] Scattering for the two-dimensional energy-critical wave equation (2009)
  • Author(s): Slim Ibrahim, Mohamed Majdoub, Nader Masmoudi, Kenji Nakanish
  • Journal Title: Duke Math.J. 150 Pages: 287-329
  • Peer Reviewed
• [Journal Article] Scattering theory for the Gross-Pitaevskii equation in three dimensions (2009)
  • Author(s): Stephen Gustafson, Kenji Nakanishi, Tai-Peng Tsai
  • Journal Title: Commun.Contemp.Math. 11 Pages: 657-707
  • Peer Reviewed
• [Presentation] Global dynamics of nonlinear dispersive equations (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: MSJ-KMS Joint Meeting 2012
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-17
• [Presentation] Global dynamicsbeyond the ground state energy fornonlinear dispersive equations (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: Journees EDP 2012
  • Place of Presentation: Domaine de Francon(フランス)
• [Presentation] Global dynamics of the3D Zakharov system (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: NonlinearHamiltonian PDEs
  • Place of Presentation: Centro StefanoFranscini (スイス)
• [Presentation] Global dynamics of the3D Zakharov system (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear PDE's @IMPA
  • Place of Presentation: Instituto Naciaonal deMatematica Pura e Aplicada (ブラジル)
• [Presentation] Global dynamics beyond the ground state energy for nonlinear dispersive equations (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: Journees EDP 2012
  • Place of Presentation: Domaine de Francon (フランス)
  • Invited
• [Presentation] Global dynamics of the 3D Zakharov system (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear Hamiltonian PDEs
  • Place of Presentation: Centro Stefano Franscini (スイス)
  • Invited
• [Presentation] Global dynamics of the 3D Zakharov system (2012)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear PDE's @ IMPA
  • Place of Presentation: Instituto Naciaonal de Matematica Pura e Aplicada (ブラジル)
  • Invited
• [Presentation] Global dynamicsabove the unstable ground state fornonlinear dispersive equations (2011)
  • Author(s): Kenji Nakanishi
  • Organizer: Asymptotic dynamics driven bysolitons and trvaling fronts innonlinear PDE
  • Place of Presentation: Santiago(チリ)
  • Year and Date: 2011-07-15
• [Presentation] Global dynamics above the unstable ground state for nonlinear dispersive equations (2011)
  • Author(s): Kenji Nakanishi
  • Organizer: Asymptotic dynamics driven by solitons and trvaling fronts in nonlinear PDE
  • Place of Presentation: Santiago, Chile(招待講演)
  • Year and Date: 2011-07-15
• [Presentation] Global dynamicsbeyond the ground energy for thefocusing nonlinear Klein-Gordonequation (2011)
  • Author(s): Kenji Nakanishi
  • Organizer: Global dynamicsbeyond the ground energy for thefocusing nonlinear Klein-Gordonequation
  • Place of Presentation: National Cheng Kung University(台湾)
  • Year and Date: 2011-01-15
• [Presentation] Global dynamics beyond the ground energy for the focusing nonlinear Klein-Gordon equation (2011)
  • Author(s): Kenji Nakanishi
  • Organizer: The 19th Workshop on Differential Equations and Its Applications
  • Place of Presentation: National Cheng Kung University台湾(招待講演)
  • Year and Date: 2011-01-15
• [Presentation] Global dynamicsbeyond the ground state energy fornonlinear Klein-Gordon equationGlobal dynamicsbeyond the ground state energy fornonlinear Klein-Gordon equation (2010)
  • Author(s): Kenji Nakanishi
  • Organizer: Harmonic analysis and PDE
  • Place of Presentation: Seoul NationalUniversity(韓国)
  • Year and Date: 2010-08-18
• [Presentation] Global dynamics beyond the ground state energy for nonlinear Klein-Gordon equation (2010)
  • Author(s): Kenji Nakanishi
  • Organizer: Harmonic analysis and PDE
  • Place of Presentation: Seoul National University大韓民国(招待講演)
  • Year and Date: 2010-08-18
• [Presentation] Scatteringthreshold for the focusing nonlinearKlein-Gordon equation (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: International Workshop onDifferential Equations and TheirApplications
  • Place of Presentation: Tainan(台湾)
  • Year and Date: 2009-12-19
• [Presentation] Scattering threshold for the focusing nonlinear Klein-Gordon equation (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: International Workshop on Differential Equations and Their Applications
  • Place of Presentation: Tainan, Taiwan
  • Year and Date: 2009-12-19
• [Presentation] Asymptoticstability and eternal oscillation ofharmonic maps in Schrodinger and heatflows (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear dispersive andgeometric evolution problems:singularities and asymptotics
  • Place of Presentation: UBC(カナダ)
  • Year and Date: 2009-08-17
• [Presentation] Asymptotic stability and eternal oscillation of harmonic maps in Schrodinger and heat flows. (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear dispersive and geometric evolution problems : singularities and asymptotics
  • Place of Presentation: UBC, Vancouver
  • Year and Date: 2009-08-17
• [Presentation] Asymptoticstability and eternal oscillation ofharmonic maps in Schrodinger and heatflows (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear PartialDifferential Equations
  • Place of Presentation: Zhangjiajie(中国)
  • Year and Date: 2009-08-10
• [Presentation] Asymptotic stability and eternal oscillation of harmonic maps in Schrodinger and heat flows (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear Partial Differential Equations
  • Place of Presentation: Zhangjiajie, China
  • Year and Date: 2009-08-10
• [Presentation] Exponential decayfor damped semilinear wave equationin the critical and the supercriticalcases (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear waves anddispersion
  • Place of Presentation: Institut Henri Poincare (フランス)
  • Year and Date: 2009-04-27
• [Presentation] Exponential decay for damped semilinear wave equation in the critical and the supercritical cases (2009)
  • Author(s): Kenji Nakanishi
  • Organizer: Nonlinear waves and dispersion
  • Place of Presentation: Institut Henri Poincare, Paris
  • Year and Date: 2009-04-27
• [Book] Invariant manifolds and dispersiveHamiltonian evolution equations (2011)
  • Author(s): Kenji Nakanishi, Wilhelm Schlag
  • Total Pages: 253
  • Publisher: European Mathematical Society
• [Book] Invariant manifolds and dispersive Hamiltonian evolution equations (2011)
  • Author(s): Kenji Nakanishi, Wilhelm Schlag
  • Total Pages: 253
  • Publisher: European Mathematical Society