Stability analysis of patterns in nonlinear dispersive equations

• Project/Area Number: 25400174
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Hiroshima University (2015-2016); Kyushu University (2013-2014)
• Principal Investigator: Mizumachi Tetsu 広島大学, 理学研究科, 教授 (60315827)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 平面孤立波 / 安定性 / 線形安定性 / レゾナンス / 安定性解析 / 平面進行波 / 長波長近似モデル / KP-II 方程式 / 平面進行波解 / line soliton / KP-II方程式 / KP-II / KdV

Outline of Final Research Achievements

I study stability of 1-line solitons for the KP-II equation and prove their stability for polynomially decaying perturbations. This is an improvement of my former result published in 2015 which shows stability of 1-line solitons for exponentially decaying perturbations. I obtain the result by splitting a small solution of the KP-II equation from the perturbations to line solitons which ensures exponential decay of the rest of the pertrubations.
I also studied the asymptotic linear stability of planar solitary wave solutions for the Benney-Luke equation which is another 2-dimensional model of long water waves.

Research Products

• [Int'l Joint Research] McMaster 大学(カナダ)
• [Int'l Joint Research] Cergy Pontoise 大学(フランス)
• [Journal Article] Stability of line solitons for the KP-II equation in R^2 (2015)
  • Author(s): Tetsu Mizumachi
  • Journal Title: Memoirs of the American Mathematical Society Volume: 238 Issue: 1125 Pages: 1-95
  • DOI: 10.1090/memo/1125
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] L^2-stability of solitary waves for the KdV equation via Pego and Weinstein's method (2014)
  • Author(s): Tetsu Mizumachi and Nikolay Tzvetkov
  • Journal Title: RIMS 講究禄別冊 Volume: B49 Pages: 33-63
  • Peer Reviewed
• [Presentation] Asymptotic linear stability of line solitary waves of the Benney-Luke equation (2016)
  • Author(s): Tetsu Mizumachi
  • Organizer: CMS Winter Meeting
  • Place of Presentation: Niagara Fall (カナダ)
  • Year and Date: 2016-12-02
  • Int'l Joint Research / Invited
• [Presentation] Stability of line solitons for the KP-II equation (2016)
  • Author(s): Tetsu Mizumachi
  • Organizer: Nonlinear Wave and Dispersive Equations
  • Place of Presentation: 京都大学
  • Year and Date: 2016-09-06
  • Int'l Joint Research / Invited
• [Presentation] Stability of line solitons for the KP-II equation (2016)
  • Author(s): Tetsu Mizumachi
  • Organizer: Nonlinear Waves 2016: May Conference
  • Place of Presentation: IHES, フランス
  • Year and Date: 2016-05-23
  • Int'l Joint Research / Invited
• [Presentation] Stability of line solitons for the KP-II equation (2016)
  • Author(s): Tetsu Mizumachi
  • Organizer: Singularity formation and long-time behavior in dispersive PDEs
  • Place of Presentation: Bonn 大学，ドイツ
  • Year and Date: 2016-03-14
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation (2016)
  • Author(s): 水町 徹
  • Organizer: 第33回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学 西新プラザ
  • Year and Date: 2016-01-27
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation (2015)
  • Author(s): 水町 徹
  • Organizer: 第2回 解析学の耳袋
  • Place of Presentation: プラサ ヴェルデ （沼津）
  • Year and Date: 2015-10-28
  • Invited
• [Presentation] On stability of line solitons of the KP-II equation (2015)
  • Author(s): 水町 徹
  • Organizer: Fundamental Problems in Mathematical and Theoretical Physics
  • Place of Presentation: 早稲田大学
  • Year and Date: 2015-09-28
  • Invited
• [Presentation] Stability of line solitons for KP-II (2015)
  • Author(s): Tetsu Mizumachi
  • Organizer: S\'{e}minaire G\'{e}om\'{e}trie, EDP et Physique Math\'{e}matique
  • Place of Presentation: Cergy-Pontoise 大学，フランス
  • Year and Date: 2015-09-21
  • Invited
• [Presentation] Stability of line solitons (2015)
  • Author(s): 水町 徹
  • Organizer: 東京大学 FMSP サマースクール『分散型方程式とソリトン』
  • Place of Presentation: 東京大学 大学院数理科学研究科
  • Year and Date: 2015-07-27
  • Invited
• [Presentation] On stability of line solitons for the KP-II equation (2015)
  • Author(s): 水町 徹
  • Organizer: RIMS研究集会「流体と気体の数学解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-07-08
  • Invited
• [Presentation] KP-II 方程式のline soliton 解の安定性 (2014)
  • Author(s): 水町 徹
  • Organizer: 日本数学会函数解析学分科会（特別講演）
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25 – 2014-09-28
  • Invited
• [Presentation] Stability of solitary waves via Pego-Weinstein's method, (2014)
  • Author(s): Tetsu Mizumachi
  • Organizer: Stability of solitary wave
  • Place of Presentation: ピサ大学, イタリア
  • Year and Date: 2014-05-26 – 2014-05-30
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation in $\mathbb {R}^2$ (2013)
  • Author(s): Tetsu Mizumachi
  • Organizer: NSF/CBMS Regional Conference in Mathematical Sciences
  • Place of Presentation: サス大学パン・アメリカン，アメリカ
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation in $\mathbb {R}^2$ (2013)
  • Author(s): Tetsu Mizumachi
  • Organizer: RIMS 研究集会「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
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation in $\mathbb {R}^2$ (2013)
  • Author(s): Tetsu Mizumachi
  • Organizer: Nonlinear Waves and Dispersive Equations
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation in $\mathbb {R}^2$ (2013)
  • Author(s): Tetsu Mizumachi
  • Organizer: Lattice Differential Equations
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach
  • Invited
• [Presentation] Stability of line solitons for the KP-II equation in $\mathbb {R}^2$ (2013)
  • Author(s): Tetsu Mizumachi
  • Organizer: SIAM Conference on Analysis of Partial Differential Equations
  • Place of Presentation: フロリダ，アメリカ
  • Invited
• [Remarks] arXiv.org > math > arXiv:1701.03390
  • URL: https://arxiv.org/abs/1701.03390
• [Remarks] arXiv.org > math > arXiv:1512.08334
  • URL: http://arxiv.org/abs/1512.08334