Research on the structure of the resonant interaction and behavior/singularity of the solutions for nonlinear dispersive wave equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K17626
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Kyoto University
• Principal Investigator: KISHIMOTO Nobu 京都大学, 数理解析研究所, 講師 (90610072)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: 非線形分散型方程式 / 初期値問題の適切性 / 周期境界条件 / 共鳴相互作用 / 解の一意性 / 解の非存在 / 組合せ論 / フーリエ制限法

Outline of Final Research Achievements

In this research, we analyzed various nonlinear dispersive wave equations including nonlinear Schrodinger equations, KdV-type equations, and even equations of rotating fluids, noticing the effect of the resonant nonlinear interaction, and obtained new results on existence, uniqueness, non-existence and other properties of solutions to the initial value problems. For a closed analysis of the resonant interaction under the periodic boundary condition we employed some tools from combinatorics, while we made the best use of existing methods such as the Fourier restriction method and the normal form transformation in accordance with each situation to control the non-resonant interaction. In particular, for some equations in which the resonant interaction becomes dominant, we proved that non-existence of solution for smooth initial data may occur, which is a totally different situation from usual dispersive equations.

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のモデルとして用いられている偏微分方程式を扱っており，また周期境界条件は数値計算を行う際に自然な設定であるため，それらの問題に対して適切性（解の一意存在および初期データの摂動に対する安定性）を厳密に証明することで，モデルを用いた数値シミュレーションの正当性の根拠が得られる．逆に解の非存在の結果は既存のモデル方程式や周期境界の設定が必ずしも適切でないことを示唆するとも考えられる．このように，適切性や非適切性の解明は，より良いモデル方程式の導出にも役立つ可能性を秘めている．