Mathematical analysis on solitary waves for nonlinear dispersive equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 22K20337
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Waseda University
• Principal Investigator: Hayashi Masayuki 早稲田大学, 理工学術院総合研究所(理工学研究所), その他(招聘研究員) (60967850)
• Project Period (FY): 2022-08-31 – 2024-03-31
• Keywords: 非線形分散型方程式 / 非線形シュレディンガー方程式 / 孤立波 / 進行波 / 代数ソリトン / 不安定性 / コーシー問題 / エネルギー空間

Outline of Final Research Achievements

We have mainly studied nonlinear dispersive equations which possess algebraically decaying solitons (algebraic solitons), and established stability/instability theory of solitary waves, constructed traveling wave solutions by variational methods, and constructed solutions in energy spaces and higher energy spaces. The research into mathematical models that can systematically handle algebraic solitons has revealed a deep connection between nonlinear analysis and linear operator theory, and the research into physical models has enabled us to discover new mathematical structures that had not been captured in previous literature.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

代数ソリトンの安定性・不安定性に関する成果は、新たな数学的知見を与えているだけでなく、代数ソリトンにまつわる数理の豊穣さを示唆しており、今後の更なる理論発展が期待される。物理モデルに対する進行波解の構成やコーシー問題の可解性の成果は、より複雑な解の大域挙動の解明や新たな物理現象の発見に繋がる可能性を秘めており、こちらも今後の発展が期待できる。