New developments in scattering theory for nonlinear dispersive equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14580
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Kagawa University (2020-2022); Tsuyama National College of Technology (2019)
• Principal Investigator: Miyazaki Hayato 香川大学, 教育学部, 准教授 (70752202)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 非線形分散型方程式 / 解挙動 / 散乱理論 / 定在波 / 安定性

Outline of Final Research Achievements

We study the behavior of solutions to nonlinear dispersive equations. In particular, we consider scattering solutions of the nonlinear Schroedinger equation. The solution behaves like linear solutions for a large time. On this research, we succeed in constructing scattering solutions that include the effects of nonlinearity in the case of that the solutions are on star graphs, or the equation has a kind of harmonic oscillator. Furthermore, we specify the condition under which standing waves of the system of nonlinear Klein-Gordon equations are strongly unstable. This result serves as a starting point for the study of scattering solutions in the system with large data.

Free Research Field

非線形偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

グラフ上の非線形偏微分方程式は、分岐構造を考慮した数理モデルを考えると自然に現れるものであり、応用上の観点からも重要な研究対象である。非線形分散型方程式の散乱理論の研究において、星グラフ上の修正散乱解の存在を示したのは初めてであり、基本的な結果といえる。また、線形部にポテンシャルを持つ場合も、 技術的な問題から空間1次元に限定した先行研究が多く、多次元において修正散乱解を構成できたことは重要な成果であったと考えている。