Analysis of resonant effects and geometric symmetry on nonlinear dispersive equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K13442
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Saga University
• Principal Investigator: Kato Takamori 佐賀大学, 理工学部, 講師 (50620639)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 非線形分散型方程式 / 初期値問題 / 適切性 / 周期境界条件 / 無条件一意性

Outline of Final Research Achievements

We considered the Cauchy problem of fifth order KdV type equations on the one-dimensional torus. We proved the well-posedness and unconditional uniqueness. This result is optimal in the sense that the nonlinear terms can be defined as the distribution. The key of this study is how to deal with the resonant parts which cannot be regarded as the perturbation of the linearized solution. By symmetry of the equation, we found that the resonant parts are exactly localized and can be cancelled by some conserved quantities.
Moreover, we obtained the improved results of fifth order modified KdV type equations and third order Benjamin-Ono type equations on the torus.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究では、実際の物理現象を記述するモデルあるいはその近似モデルとなる偏微分方程式を扱う。実際の現象は様々な設定で考察されるが、数値計算を実行する上で周期境界条件は最も自然な設定である。そのため、周期境界条件下で偏微分方程式に対する適切性(解の一意存在及び初期値に関する連続依存性)及び非適切性を厳密に示すことは、数値シミュレーションの正当性やモデルとなる方程式と実際の現象との整合性を判定する際に大きな役割を担う。