An investigation of symmetries in the geometric structure and existence of global solutions to nonlinear dispersive wave equations

2017 Fiscal Year Final Research Report

• Project/Area Number: 25287022
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Kobe University (2017); Hokkaido University (2013-2016)
• Principal Investigator: Takaoka Hideo 神戸大学, 理学研究科, 教授 (10322794)
• Co-Investigator(Renkei-kenkyūsha): KUBO Hideo 北海道大学, 大学院理学研究院, 教授 (50283346) ; NAKANISHI Kenji 京都大学, 数理解析研究所, 教授 (40322200) ; TSUGAWA Kotaro 中央大学, 大学院理工学研究科, 教授 (70402451)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 分散型方程式 / 適切性 / 非線形波動

Outline of Final Research Achievements

In this study, I have developed the local and global well-posedness for the initial value problem related to the nonlinear Schrodinger equations in which dispersion effect and nonlinear interaction effect are incorporating. Using the Fourier analysis, I separated the solution into two parts; non-resonant and resonant oscillation parts, which have different in nature and distinguish nonuniformity part of solutions. For the nonlinear Schrodinger equations both with derivative in nonlinearities and on a sphere domain, I improved the local well-posedness for large function spaces. Moreover, I showed that there exists exchange of energy between Fourier modes. In the research process, I observed the estimation of energy exchange between different Fourier modes, due to the contribution in the nonlinear interaction.

Free Research Field

偏微分方程式