Stability analysis for planar traveling solutions for nonlinear dispersive equations

• Project/Area Number: 17K05332
• 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
• Principal Investigator: Mizumachi Tetsu 広島大学, 先進理工系科学研究科(理), 教授 (60315827)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: Benney Luke 方程式 / 線状孤立波解 / 安定性 / 長波長近似モデル / KP-II方程式 / 1-線ソリトン解 / 平面孤立波解 / 多線ソリトン / 線形安定性 / ２次元浅水波モデル / 線状孤立波 / 線ソリトン / 位相の時間発展 / 非線形分散型方程式 / 平面進行波 / 安定性解析

Outline of Final Research Achievements

I prove transverse stability of line solitary wave solutions for the Benney-Luke equations which is a long wave model for 3-dimensional water waves. The Benney-Luke is an isotropic model whereas the KP-II equation is a unidirectional model. Nevertheless, it turns out that perturbations to a line solitary wave propagate along its crest in the same manner as those for the KP-II equation. I also prove that phase shift of modulating line solitary waves for the KP-II equation and the Benney-Luke equation remains small for all the time.

Academic Significance and Societal Importance of the Research Achievements

３次元水面波の長波長近似モデルの線状孤立波解の全空間における安定性は，完全可積分系の方程式であるKP-II方程式の場合に知られていたが，完全可積分系でないBenney-Luke方程式に対しても同様の結果を得ることが出来た．

Research Products

• [Int'l Joint Research] Academia Sinica(台湾)
• [Int'l Joint Research] Carnegie Mellon 大学(米国)
• [Journal Article] Stability of Benney--Luke Line Solitary Waves in 2 Dimensions (2020)
  • Author(s): Mizumachi Tetsu、Shimabukuro Yusuke
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 52 Issue: 5 Pages: 4238-4283
  • DOI: 10.1137/19m1253848
  • Peer Reviewed
• [Journal Article] The Phase Shift of Line Solitons for the KP-II Equation (2019)
  • Author(s): Mizumachi Tetsu
  • Journal Title: Fields Institute Communications Volume: 83 Pages: 433-495
  • DOI: 10.1007/978-1-4939-9806-7_10
  • ISBN: 9781493998050, 9781493998067
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Stability of line solitons for the KP-II equation in R^2. II. (2018)
  • Author(s): Tetsu Mizumachi
  • Journal Title: Proc. Roy. Soc. Edinburgh Sect. A Volume: 148 Issue: 1 Pages: 149-198
  • DOI: 10.1017/s0308210517000166
  • Peer Reviewed
• [Journal Article] Asymptotic linear stability of Benney-Luke line solitary waves in 2D (2017)
  • Author(s): Tetsu Mizumachi and Yusuke Shimabukuro
  • Journal Title: Nonlinearity Volume: 30 Issue: 9 Pages: 3419-3465
  • DOI: 10.1088/1361-6544/aa7cc7
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Stability of line solitary waves for some long wave models (2019)
  • Author(s): 水町 徹
  • Organizer: 2019 Workshop on Nonlinear Dispersive Partial Differential Equations and Inverse Scattering
  • Int'l Joint Research / Invited
• [Presentation] On stability of line solitary waves for long wave models (2019)
  • Author(s): 水町 徹
  • Organizer: 第36回九州における偏微分方程式研究集会
  • Int'l Joint Research / Invited
• [Presentation] On the phase shift of line solitary waves for the KP-II equation (2018)
  • Author(s): 水町 徹
  • Organizer: Workshop on Nonlinear Water Waves
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic Linear Stability of Benney-Luke line solitary waves (2017)
  • Author(s): Tetsu Mizumachi
  • Organizer: Workshop on Inverse Scattering and Dispersive PDEs in Two Space Dimensions
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic Linear Stability of Benney-Luke line solitary waves (2017)
  • Author(s): Tetsu Mizumachi
  • Organizer: Tosio Kato Centennial Conference
  • Int'l Joint Research / Invited
• [Remarks] Stability of Benney-Luke line solitary waves in 2D
  • URL: https://arxiv.org/abs/1904.01142
• [Remarks] The phase shift of line solitons for the KP-II
  • URL: https://arxiv.org/abs/1808.00809