Global studies on solitary waves for nonlinear dispersive equations

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03379
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokyo University of Science
• Principal Investigator: OHTA Masahito 東京理科大学, 理学部第一部数学科, 教授 (00291394)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Keywords: 非線形シュレディンガー方程式 / 孤立波 / 安定性

Outline of Final Research Achievements

We proved the strong instability of standing waves for a nonlinear Schroedinger eqution with an inverse power potential including the Coulomb potential in multi-dimensional space, which can be regarded as a generalization of a nonlinear Schroedinger equation with a delta-function potential in one space dimension.
Moreover, we studied the stability of solitary waves for a nonlinear Klein-Gordon equation with a delta-function potential, a generalized Boussinesq equation, a nonlinear Schroedinger equation on a star graph.

Free Research Field

非線形偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

非線形シュレディンガー方程式や非線形クライン・ゴルドン方程式の孤立波解の安定性と不安定性を通して、非線形分散型方程式の解の大域的漸近挙動に関する新しい知見を得ることができた。
特に、さまざまなタイプの相互作用を含む非線形シュレディンガー方程式の定在波解の強い意味での不安定性を示すことにより、孤立波解の不安定性と爆発解の関係について明らかにすることができた。