Dynamics of solutions for nonlinear dispersive equation

• Project/Area Number: 21540220
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kyushu University
• Principal Investigator: MIZUMACHI Tetsu 九州大学, 大学院・数理学研究院, 准教授 (60315827)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 孤立波 / 安定性 / 非線形分散型方程式 / 平面進行波 / KP方程式 / KP-II方程式 / line-soliton / 1次元Benney-Luke / ソリトンの安定性 / KP-II / L^2安定性 / 一次元非線形シュレディンガー方程式 / Fermi-Pasta-Ulamの格子模型 / 多ソリトン

Research Abstract

I study stability of solitary waves for long wave models using their monotonicity preoperty. I show that modulations of the line soliton for the KP-II equation can be described a system of the Burgers equation and prove the stability of line soliton solution. I also prove stability of multi solitary waves of the Fermi-Pasta-Ulam lattices in the KdV limit and solitary wave solutions of the Benney-Luke equation which is a kind of bidirectional models akin to the Boussinesq systems. In a joint work with Tzvetkov, we show that stability of line soliton solutions for the KP-II equation in L^2(R_x×T_y) is equivalent to the stability of the null solution by using the Miura transformation and obtain L^2(R_x×T_y) stability of line solitons. The idea was tranfered to 1D cubic NLS in a joint work with Pelinovsky.

Research Products

• [Journal Article] Asymptotic stabilityof N-solitons of the FPU lattices (2013)
  • Author(s): T. Mizumachi
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: 207巻2号 Issue: 2 Pages: 393-457
  • DOI: 10.1007/s00205-012-0564-x
  • Peer Reviewed
• [Journal Article] Asymptotic stability of solitary waves in the Benney-Luke model of water waves (2013)
  • Author(s): Tetsu Mizumachi, Jos'e Ra'ul Quintero and Robert L. Pego
  • Journal Title: Differential and Integral Equations Volume: 26巻3-4号 Pages: 253-301
  • Peer Reviewed
• [Journal Article] Stability of the line soliton of the KP-II equation under periodic transverse perturbations (2012)
  • Author(s): T. Mizumachi and N.Tzvetkov
  • Journal Title: Mathematische Annalen Volume: 352巻3号 Issue: 3 Pages: 659-690
  • DOI: 10.1007/s00208-011-0654-3
  • Peer Reviewed
• [Journal Article] On the asymptotic stability of localized modes in the discrete nonlinear Schroedinger equation (2012)
  • Author(s): T.Mizumachi, D.Pelinovsky
  • Journal Title: Discrete and Continuous Dynamical Systems, Series S Volume: 5(5) Issue: 5 Pages: 971-987
  • DOI: 10.3934/dcdss.2012.5.971
  • Peer Reviewed
• [Journal Article] B\"{a}cklund transformation and $L^2$-stability of NLS solitons (2012)
  • Author(s): T.Mizumachi, D.Pelinovsky
  • Journal Title: International Mathematics Research Notices Volume: 2012(9) Pages: 2034-2067
  • DOI: 10.1093/imrn/rnr073
  • Peer Reviewed
• [Journal Article] N-soliton states of the FPU lattices (2011)
  • Author(s): T.Mizumachi
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 43(5) Issue: 5 Pages: 2170-2210
  • DOI: 10.1137/100792457
  • Peer Reviewed
• [Journal Article] Description of the inelastic collision of two solitary waves for the BBM equation (2010)
  • Author(s): Y. Martel, F. Merle and T. Mizumachi
  • Journal Title: Archive for Rational Mechanics and Analysis Volume: 196巻2号 Issue: 2 Pages: 517-574
  • DOI: 10.1007/s00205-009-0244-7
  • Peer Reviewed
• [Journal Article] Description of the inelastic collision of two solitary waves for the BBM equation (2010)
  • Author(s): Frank Merle, Yvan Martel, Tetsu Mizumachi
  • Journal Title: Archive for Rational Mechanics and Analysis online first
  • Peer Reviewed
• [Journal Article] Asymptotic stability of lattice solitons in the energy space (2009)
  • Author(s): T. Mizumachi
  • Journal Title: Communicaitons in Mathematical Physics Volume: 288巻1号 Issue: 1 Pages: 125-144
  • DOI: 10.1007/s00220-009-0768-6
  • Peer Reviewed
• [Journal Article] Asymptotic stability of lattice solitons in the energy space (2009)
  • Author(s): Tetsu Mizumachi
  • Journal Title: Communications in Mathematical Physics 288(1) Pages: 125-144
  • Peer Reviewed
• [Presentation] On stability of line solitons of KP-II (2012)
  • Author(s): 水町 徹
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学
  • Year and Date: 2012-12-16
• [Presentation] Asymptotic stability of solitary waves in the Benney-Luke model of water waves (2012)
  • Author(s): Tetsu Mizumachi
  • Organizer: Evolution Equations: A Workshop in Honor of Terence Tao
  • Place of Presentation: Northwestern University
  • Invited
• [Presentation] On stability of line solitons of KP-II (2012)
  • Author(s): Tetsu Mizumachi
  • Organizer: Analytical and Numerical Advances Around Schroedinger Equations
  • Place of Presentation: Institut de Mathematiques de Toulouse
  • Invited
• [Presentation] BacklundTransformation and L2-stability of NLS Solitons (2011)
  • Author(s): 水町 徹
  • Organizer: International Congress on Industrial and Applied Mathematics(Contributed Minisymposia Dispersive Equations in Mathematical Physics)
  • Place of Presentation: カナダ
  • Year and Date: 2011-07-20
• [Presentation] B\"{a}cklund transformation and $L^2$-stability of NLS Solitons (2011)
  • Author(s): 水町徹
  • Organizer: ICIAM2011
  • Place of Presentation: カナダ,バンクーバー
  • Year and Date: 2011-07-20
• [Presentation] Stability of the Line Soliton of the KP-II Equation in L2(R_x×T_y) (2011)
  • Author(s): 水町 徹
  • Organizer: SIAM Conference on Applications of Dynamical Systems
  • Place of Presentation: Snowbird
  • Year and Date: 2011-05-22
• [Presentation] Stability of the Line Soliton of the KP-II Equation in L^2(R_x×T_y) (2011)
  • Author(s): 水町徹
  • Organizer: SIAM Conference on Dynamical Systems
  • Place of Presentation: アメリカ,Snowbird(招待講演)
  • Year and Date: 2011-05-22
• [Presentation] Miura transformation and orbital stability of KP-II solitons (2011)
  • Author(s): 水町徹
  • Organizer: IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena
  • Place of Presentation: アメリカ,ジョージア大学(招待講演)
  • Year and Date: 2011-04-04
• [Presentation] Stability of co-propagating N-solitons of FPU lattices in the KdV limit (2009)
  • Author(s): 水町 徹
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Place of Presentation: アメリカ合衆国マイアミ
  • Year and Date: 2009-12-09
• [Remarks] KP-II方程式のline soliton解の安定性に関する最新の成果を公開した.
  • URL: http://arxiv.org/abs/1303.3532
• [Remarks] Stability of line solitons for KP-II in R^2
  • URL: http://arxiv.org/pdf/1303.3532v1.pdf