Study on oscillation and singularity in mass resonance

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610027
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Waseda University
• Principal Investigator: Tohru Ozawa 早稲田大学, 理工学術院, 教授 (70204196)
• Co-Investigator(Kenkyū-buntansha): TANAKA Kazunaga 早稲田大学, 理工学術院, 教授 (20188288)
• Co-Investigator(Renkei-kenkyūsha): KATAYAMA Soichiro 和歌山大学, 教育学部, 教授 (70283942) ; SUNAGAWA Hideaki 大阪大学, 理学(系)研究科(研究院), 准教授 (80375394) ; MACHIHARA Shuji 埼玉大学, 教育学部, 准教授 (20346373)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 関数方程式の大域理論

Outline of Final Research Achievements

Mass resonance phenomenon of systems of nonlinear Schr\"odinger equations is studied. We have found an appropriate Lagrangean for the system of Schr\"odinger equations with quadratic interactions, by which we are able to give a clear explanation of the roles of mass resonance phenomenon as well as of complex conjugate in wavefunctions.
A variational setting has been introduced for the analysis on standing waves associated with mass resonance. We proved the existence and (essential) uniqueness of ground states.
Moreover, examples of interactions were found for the blow-up solutions with arbitrarily small Cauchy data under mass resonance condition.

Free Research Field

数物系科学