On the behavior of solutions to some nonlinear Schrodinger equations

Research Project

Project/Area Number	25400178
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Research Field	Mathematical analysis
Research Institution	Kumamoto University (2015) University of Miyazaki (2013-2014)
Principal Investigator	Kita Naoyasu 熊本大学, 自然科学研究科, 教授 (70336056)
Project Period (FY)	2013-04-01 – 2016-03-31
Project Status	Completed (Fiscal Year 2015)
Budget Amount *help	¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000) Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000) Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords	非線形シュレディンガー方程式 / 解の漸近挙動 / 解の爆発 / 非線形消散項 / 非線形増幅項 / 解の減衰評価 / 漸近挙動
Outline of Final Research Achievements	I have reserched the asymptotic behaviors and blowing-up phenomena of solutions to the Cauchy problem for some nonlinear Schrodinger equations (NLS) which including complex coefficient in the nonlinearity. This kind of NLS describes shape change of puls - electro-magnetic wave - propagating through optical fibers. it is classified with the two cases - (1)nonlinear energy-dissipation and (2) nonlinear amplification. As results of my reserch supported by this national grant, in case (1), I specified the decay order of the solutions in the uniform norm even though the initial datum are attained without size-restriction. In addition, I could prove that the solutions are gradually minimized in L^2 norm as well. In case (2), the L^2-norm of the solutions blow up in finite time for small initial data.