Elucidation of the asymptotic behavior of the solution to nonlinear dispersive equations in high dimensions

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14578
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Okayama University of Science
• Principal Investigator: Uriya Kota 岡山理科大学, 理学部, 講師 (10779474)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 分散型方程式 / 漸近挙動 / 散乱理論

Outline of Final Research Achievements

As a generalization of nonlinear Schroedinger equation with power type nonlinearities, we studied the final state problem for the inhomogeneous nonlinear Schroedinger equation. As a byproduct, we obtain the asymptotic behavior of the solution to the inhomogeneous nonlinear Schroedinger equation with inverse square potential in high dimensions.
We also studied the asymptotic behavior of the solution to cubic nonlinear Klein-Gordon system/nonlinear Schroedinger system in one dimension, nonlocal nonlinear Schroedinger equation, 4th order derivative Schroedinger equation, nonlinear Schroedinger equation on star-graph.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

非斉次非線形シュレディンガー方程式の解の漸近挙動の研究により，高次元における非線形分散型方程式の解の漸近挙動が得られる一つのモデルを与えることができた．特殊な例かもしれないが，高次元の解の漸近挙動を解明するための端緒となることが期待される．また，1次元3次の非線形クライン-ゴルドン方程式系や非線形シュレディンガー方程式系の解の漸近挙動の分類は類似の構造を持つ非線形偏微分方程式系の様々な研究に応用が可能なものである．