Research for decay and blow-up of solutions to nonlinear Schrodinger equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05305
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kumamoto University
• Principal Investigator: Kita Naoyasu 熊本大学, 大学院先端科学研究部(工), 教授 (70336056)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 非線型分散型方程式 / 解の減衰評価 / 解の漸近挙動 / 非線型シュレーディンガー方程式

Outline of Final Research Achievements

In this research, we considered decay rates of the solutions to dissipative nonlinear Schrodinger equations. A nonlinear Schrodinger equation modes the evolution of pulses (or electro-magnetic waves) propagating through optical fibers. In particular, the model treated in this research describes how a pulse is weaken by the impurities lying the optical fiber. We could reveal that, if the power of nonlinearity is Barab-Ozawa’s critical (or sub-critical), the L∞-norm of the solution decays like t{-1/2} (log t){-1/2}, and the L2-norm decays dependently on the regularity of the data but the decay rate tends to (log t){-1/2} as the regularity of the data is refined. In our research, the optimality of these decay rates is also proved. The word “optimality” means that, if a solution decays more rapidly than t{-1/2} (log t){-1/2} in L∞ (or (log t){-1/2} in Ｌ2), then the solution must be trivial (identically equal to 0).

Free Research Field

非線型偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

【産業的な意義】光ファイバーの中を伝わる信号が何km伝わるごとにその強さが半分になるのか，定量的に算出することができた。これは，信号増幅器を何Kmおきに設置すれば良いのか見積もりができるという点で意義がある。数値シミュレーションで信号の減衰を予測する場合には，差分化の精度やプログラムの安定性が懸案になるため，結果の信頼性に疑問が付きまとう。しかし，数学的な解析によって得られた結果には，そのような不備が無いところに利点がある。