Research on minimization problem related with stabilizing effect for standing waves caused by non-local interaction

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03383
• 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: Kyoto Sangyo University
• Principal Investigator: Watanabe Tatsuya 京都産業大学, 理学部, 教授 (60549749)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 楕円型微分方程式 / 変分問題 / 定在波 / 安定性解析

Outline of Final Research Achievements

In this research, I have investigated minimization problems related with the stabilizing effect for standing waves caused by non-local interactions. Especially, I have studied the solution structure (uniqueness, multiplicity and asymptotic profile) of the stationary problem of a quasilinear Schroedinger equation which is derived as a limit equation. One of the main results of my research is to show the uniqueness and the non-degeneracy of ground state solutions for a whole range of parameters. Moreover, I have studied the asymptotic behavior of solutions when the nonlinear term has the Sobolev critical growth. My result completely reveals how the parameter dependence of the asymptotic profiles of solutions changes with respect to the nonlinear exponent.
I have also performed the stability analysis of standing waves for the Schroedinger-Maxwell system, and investigated several other physical models.

Free Research Field

微分方程式と変分問題

Academic Significance and Societal Importance of the Research Achievements

本研究では、定在波の安定化効果が非局所的相互作用によって得られるということを、付随する最小化問題のラグランジュ乗数の解析から捉えることを目標としている。定在波は様々な数理モデルで現れ、その安定性解析は重要な研究課題の一つであるが、厳密な解析が行われていない数理モデルは数多く残されている。本研究成果で得られた結果や手法は他の方程式に対する最小化変分問題にも応用できる。様々な物理モデルにおける定在波の安定性解析の研究にも大きな寄与があり、物理学者による(形式的な)考察を数学的に厳密化でき、応用面でも大きな意義がある。