An investigation of the continuation problem for the nonlinear Schroedinger equations beyond the singularity

2014 Fiscal Year Final Research Report

• Project/Area Number: 23654052
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Meiji University (2013-2014); Osaka University (2011-2012)
• Principal Investigator: NAWA Hayato 明治大学, 理工学部, 教授 (90218066)
• Co-Investigator(Kenkyū-buntansha): ISHIWATA Tetsuya 芝浦工業大学, システム工学部, 教授 (50334917)
• Research Collaborator: FIBICH Gadi Tel Aviv University, 教授 ; SULEM Catherine University Tronto, 教授
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 非線形シュレーディンガー方程式 / 爆発解 / ネルソン拡散過程 / 曲線の運動 / クリスタライン / 等位面 / 爆発速度

Outline of Final Research Achievements

This project was devoted to the study of the continuation problem of blowup solutions of the pseudo-conformally invariant nonlinear Schroedinger equation beyond the singularity. We tried to establish a rigorous mathematical concept to extend blowup solutions beyond the blowup time so that we investigated the precise behavior of the solution near the blowup time by means of the Nelson diffusion as well as a system of degenerate ordinary differential equations which had been found during the preparation of this project: the system of ODE describes the behavior of points in a level set of the square of the absolute value of the solution. Although we could not succeed to establish a significant concept of extending the blowup solutions, we obtained an estimate on the blowup rate for a class of blowup solutions and learned the importance of the numerical study to get an idea which lead us to a new KAKENHI project investing the blowup problem including other type of evolutional equations.

Free Research Field

非線形偏微分方程式と応用確率解析