Asymptotic analysis of nonlinear dispersive equations with critical nonlinearities

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03680
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tohoku University
• Principal Investigator: Hayashi Nakao 東北大学, 理学研究科, 特任教授 (30173016)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 非線形シュレディンガー方程式 / 臨界べき非線形項 / 解の漸近的振る舞い / 非斉次境界値問題 / 散乱問題 / 非線形境界値問題 / 自己相似解

Outline of Final Research Achievements

We studied asymptotic behavior of solutions to fractional nonlinear Schrodinger equations and fractional Korteweg-de Vries equations. The effect of nonlinearities to the asymptotic behavior of solutions was shown. We also studied nonlinear dispersive equations satisfying mass conservation law and the solutions were stability in the neighborhood of self-similar solutions. In these studies, we decomposed the evolution operators into some operators and consider each operator carefully to apply the proofs of our results. This method is called the factorization technique and used in the study of nonlinear dispersive equations. Initial boundary value problem for nonlinear Schrodinger equations was considered and global existence in time of solutions was shown in a scale invariant space.

Free Research Field

Nonlinear dispersive equations

Academic Significance and Societal Importance of the Research Achievements

流体力学の研究、量子力学の研究で用いられる方程式の多くは、非線形項が臨界冪である方程式であり、非線形項、初期値が解の性質に影響を与えることが知られている。一方、解の漸近的振る舞いにこれらがどの様に現れるかなど、明らかにされていない点も多い。我々は発展作用素に対する因数分解公式の方法を用いて、臨界べき非線形シュレディンガー方程式の解の漸近的振る舞いを明らかにした。この方法がより廣い非線形分散型波動方程式に応用可能であることを擬微分作用素の有界性定理を用いた研究成果で示したことは学術的意義であると考える。